4tx codebook enhancement in lte

ABSTRACT

Channel state information (CSI) feedback in a wireless communication system is disclosed. A precoding matrix is generated for multi-antenna transmission based on precoding matrix indicator (PMI) feedback, wherein the PMI indicates a choice of precoding matrix derived from a matrix multiplication of two matrices from a first codebook and a second codebook. In one embodiment, the first codebook comprises at least a first precoding matrix constructed with a first group of adjacent Discrete-Fourier-Transform (DFT) vectors. In another embodiment, the first codebook comprises at least a second precoding matrix constructed with a second group of uniformly distributed non-adjacent DFT vectors. In yet another embodiment, the first codebook comprises at least a first precoding matrix and a second precoding matrix, where said first precoding matrix is constructed with a first group of adjacent DFT vectors, and said second precoding matrix is constructed with a second group of uniformly distributed non-adjacent DFT vectors.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a Continuation of application Ser. No. 14/177,547filed Feb. 11, 2014, which claims priority under 35 U.S.C. 119(e)(1) toU.S. Provisional Patent Application No. 61/763,804, filed on Feb. 12,2013; U.S. Provisional Patent Application No. 61/768,851, filed on Feb.25, 2013; U.S. Provisional Patent Application No. 61/769,463, filed onFeb. 26, 2013; U.S. Provisional Patent Application No. 61/770,705, filedon Feb. 28, 2013; U.S. Provisional Patent Application No. 61/771,664,filed on Mar. 12, 2013; U.S. Provisional Patent Application No.61/807,647, filed on Apr. 2, 2013; U.S. Provisional Patent ApplicationNo. 61/812,459, filed on Apr. 16, 2013; and U.S. Provisional PatentApplication No. 61/817,657, filed on Apr. 30, 2013, all titled “4TxCodebook Enhancement in Release 12,” the disclosures of which are herebyincorporated by reference herein in their entirety.

TECHNICAL FIELD

The technical field of this invention is wireless communication such aswireless telephony.

BACKGROUND

The present embodiments relate to wireless communication systems and,more particularly, to the precoding of Physical Downlink Shared Channel(PDSCH) data and dedicated reference signals with codebook-basedfeedback for multi-input multi-output (MIMO) transmissions.

With Orthogonal Frequency Division Multiplexing (OFDM), multiple symbolsare transmitted on multiple carriers that are spaced apart to provideorthogonality. An OFDM modulator typically takes data symbols into aserial-to-parallel converter, and the output of the serial-to-parallelconverter is considered as frequency domain data symbols. The frequencydomain tones at either edge of the band may be set to zero and arecalled guard tones. These guard tones allow the OFDM signal to fit intoan appropriate spectral mask. Some of the frequency domain tones are setto values which will be known at the receiver. Among these are ChannelState Information Reference Signals (CSI-RS) and Dedicated orDemodulating Reference Signals (DMRS). These reference signals areuseful for channel estimation at the receiver.

In multi-input multi-output (MIMO) communication systems with multipletransmit/receive antennas, the data transmission is performed viaprecoding. Here, precoding refers to a linear (matrix) transformation ofL-stream data into P-stream where L denotes the number of layers (alsotermed the transmission rank) and P denotes the number of transmitantennas. With the use of dedicated (i.e. user-specific) DMRS, atransmitter, such as a base station or eNodeB (eNodeB), can performprecoding operations that are transparent to user equipment (UE) actingas receivers. It is beneficial for the base station to obtain aprecoding matrix recommendation from the user equipment. This isparticularly the case for frequency-division duplexing (FDD) where theuplink and downlink channels occupy different parts of the frequencybands, i.e. the uplink and downlink are not reciprocal. Hence, acodebook-based feedback from the UE to the eNodeB is preferred. Toenable a codebook-based feedback, a precoding codebook needs to bedesigned.

The Long-Term Evolution (LTE) specification includes codebooks for2-antenna, 4-antenna, and 8-antenna transmissions. While those codebooksare designed efficiently, the present inventors recognize that stillfurther improvements in downlink (DL) spectral efficiency are possible.Accordingly, the preferred embodiments described below are directedtoward these problems as well as improving upon the prior art.

SUMMARY

Systems and methods for channel state information (CSI) and precodingmatrix indicator (PMI) feedback in a wireless communication system aredisclosed. A precoding matrix is generated for multi-antennatransmission based on a precoding matrix indicator (PMI) feedback fromat least one remote receiver wherein the PMI indicates a choice ofprecoding matrix derived from a matrix multiplication of two matricesfrom a first codebook and a second codebook. One or more layers of adata stream are precoded with the precoding matrix and transmitted tothe remote receiver.

In one embodiment, a method of CSI feedback and transmitting data in awireless communication system comprises receiving one or more precodingmatrix indicator (PMI) signals from a remote transceiver. The PMIsignals indicating a choice of a precoding matrix W. The systemgenerates the precoding matrix W from a matrix multiplication of twomatrices W₁ and W₂. Matrix W is termed the composite precoder. Matrix W1targets wideband/long-term channel properties, and matrix W2 targetsfrequency-selective/short-term channel properties. Each of thecomponents W1, W2 is assigned a codebook. Hence, two distinct codebooksare needed: C₁ and C₂. Matrix W₁ is selected from a first codebook C₁based on a first group of bits in the PMI signals, and matrix W₂ isselected from a second codebook C₂ based on a second group of bits inthe PMI signals.

Proposed first and second codebooks C₁ and C₂ are defined below fordifferent ranks and different PMI bit lengths.

In one embodiment, one or more layers of a data stream are precoded bymultiplication with the precoding matrix W. The precoded layers in thedata stream are then transmitted to the remote receiver.

BRIEF DESCRIPTION OF THE DRAWINGS

These and other aspects of this invention are illustrated in thedrawings, in which:

FIG. 1 illustrates an exemplary wireless telecommunications network.

FIG. 2 illustrates a uniform linear array (ULA) or four pairs of ULAelements.

FIG. 3 illustrates four pairs of cross-polarized arrays.

FIG. 4 illustrates an example in which a grid of beams is shifted by m=2beams.

FIG. 5 illustrates an example of using a row selection vector to pruneantenna pairs in an 8Tx array.

FIG. 6 illustrates the technique used in downlink LTE-Advanced (LTE-A).

FIG. 7 is a block diagram illustrating internal details of a mobile UEand an eNodeB in an exemplary network system.

DETAILED DESCRIPTION

The invention(s) now will be described more fully hereinafter withreference to the accompanying drawings. The invention(s) may, however,be embodied in many different forms and should not be construed aslimited to the embodiments set forth herein. Rather, these embodimentsare provided so that this disclosure will be thorough and complete, andwill fully convey the scope of the invention(s) to a person of ordinaryskill in the art. A person of ordinary skill in the art may be able touse the various embodiments of the invention(s).

FIG. 1 illustrates an exemplary wireless telecommunications network 100.Network 100 includes a plurality of base stations 101, 102 and 103. Inoperation, a telecommunications network necessarily includes many morebase stations. Each base station 101, 102 and 103 (eNodeB) is operableover corresponding coverage areas 104, 105 and 106. Each base station'scoverage area is further divided into cells. In the illustrated network,each base station's coverage area is divided into three cells 104 a-c,105 a-c, 106 a-c. User equipment (UE) 107, such as telephone handset, isshown in Cell A 104 a. Cell A 104 a is within coverage area 104 of basestation 101. Base station 101 transmits to and receives transmissionsfrom UE 107. As UE 107 moves out of Cell A 104 a and into Cell B 105 b,UE 107 may be handed over to base station 102. Because UE 107 issynchronized with base station 101, UE 107 can employ non-synchronizedrandom access to initiate handover to base station 102.

Non-synchronized UE 107 also employs non-synchronous random access torequest allocation of uplink 108 time or frequency or code resources. IfUE 107 has data ready for transmission, which may be traffic data,measurements report, tracking area update, UE 107 can transmit a randomaccess signal on uplink 108. The random access signal notifies basestation 101 that UE 107 requires uplink resources to transmit the UEsdata. Base station 101 responds by transmitting to UE 107 via downlink109, a message containing the parameters of the resources allocated forUE 107 uplink transmission along with a possible timing errorcorrection. After receiving the resource allocation and a possibletiming advance message transmitted on downlink 109 by base station 101,UE 107 optionally adjusts its transmit timing and transmits the data onuplink 108 employing the allotted resources during the prescribed timeinterval.

Base station 101 configures UE 107 for periodic uplink soundingreference signal (SRS) transmission. Base station 101 estimates uplinkchannel quality information (CSI) from the SRS transmission. Thepreferred embodiments of the present invention provide improvedcommunication through precoded multi-antenna transmission withcodebook-based feedback. In a cellular communication system, a UE isuniquely connected to and served by a single cellular base station oreNodeB at a given time. An example of such a system is the 3GPP LTEsystem, which includes the LTE-Advanced (LTE-A) system. With anincreasing number of transmit antennas at the eNodeB, the task ofdesigning an efficient codebook with desirable properties ischallenging.

CSI consists of Channel Quality Indicator (CQI), precoding matrixindicator (PMI), precoding type indicator (PIT), and/or rank indication(RI). The time and frequency resources that can be used by the UE toreport CSI are controlled by the eNodeB.

In one embodiment, a dual-stage codebook for CSI feedback is based onthe product structure proposed in:

W=W ₁ W ₂  (1)

where W₁ targets wideband/long-term channel properties and W₂ targetsfrequency-selective/short-term channel properties. Each of thecomponents W₁, W₂ is assigned a codebook. Hence, two distinct codebooksare needed: CB₁ and CB₂. W is termed the composite precoder. The choiceof W₁ and W₂ are indicated via PMI₁ and PMI₂.

The following principles are enforced for codebook designs:

(1) Finite alphabet for W: each matrix element belongs to a finite setof values or constellation (e.g., M-PSK alphabet).

(2) Constant modulus for W: all elements in a precoding matrix have thesame magnitude. This is important to facilitate power amplifier (PA)balance property in all scenarios. Note that constant modulus is asufficient condition for PA balance, but not a necessary condition.However, enforcing constant modulus property tends to result in asimpler codebook design. Note also that while the precoding codebook(for feedback) conforms to the constant modulus property, this does notrestrict the eNodeB from using non-constant modulus precoder. This ispossible due to the use of UE-specific RS for demodulation.

(3) Nested property for W: every matrix/vector of rank-n is a sub-matrixof a rank-(n+1) precoding matrix, n=1, 2, . . . , N−1 where N is themaximum number of layers. While this property is desirable as it allowsto reduce the complexity of PMI selection, it is not necessary tofacilitate rank override if UE-specific RS is used.

(4) The associated feedback signaling overhead should be minimized. Thisis achieved by a balance between the overhead associated with W₁(wideband, long-term) and W₂ (sub-band, short-term). Here, both the time(feedback rate) and frequency (feedback granularity) dimensions areimportant.

-   -   (a) Blindly increasing the size of CB₁ (while reducing the size        of CB₂) does not guarantee reducing the overall feedback        overhead if a certain level of performance is expected. If the        codebook CB₁ is meant to cover a certain precoder sub-space with        a given spatial resolution, increasing the size of CB₁ demands        an increase in feedbacks signaling associated with W₁, both in        time and frequency. This is because CB₁ starts to capture        shorter-term channel properties, which are meant to be parts of        CB₂.    -   (b) To ensure that CB₁ does not need to be updated too        frequently (in time and frequency), CB₁ should capture long-term        channel properties such as the antenna setup and a range of        values of angle of departure (AoD) which are associated with        spatial correlation.    -   (c) The design should strive to keep the maximum overhead        associated with W₂/CB₂ the same as the Release 8 PMI overhead        (i.e., ≦4 bits).

(5) Unitary precoder for W: (the column vectors of a precoder matrixmust be pair-wise orthogonal to one another), while not necessary, is asufficient condition for maintaining constant average transmitted power.This constraint is also used in designing the codebook at least for somerelevant ranks.

The design of a 4Tx codebook for LTE disclosed herein targets anenhancement for multi-user (MU) MIMO over the 4Tx codebook in LTERelease 8. Rather than redesigning the 4Tx codebook, the enhancementsdisclosed herein focus on enhancing the MU-MIMO performance because theLTE Release 8 4Tx codebook was already designed to offer competitiveperformance for single-user (SU) MIMO (while keeping MU-MIMO in mindwith the support of 8 discrete Fourier transform (DFT) vectors in therank-1 codebook). Based on this consideration, the 4Tx enhancementfocuses on rank-1 and at most rank-2 where MU-MIMO becomes relevant.

In terms of antenna setups, three setups may be considered:

two dual-polarized elements with λ/2 (half wavelength) spacing betweentwo elements,

two dual-polarized elements with 4λ (larger) spacing between twoelements, and

uniform linear array (ULA) with λ/2 (half wavelength) spacing.

The first and the second setups have the highest priority. Goodperformance should be ensured for dual (i.e., cross) polarized antennaarrays with both small and large spacing.

The antenna element indexing shown in FIGS. 2 and 3 is use enumerate thespatial channel coefficients H_(n,m), where n and m are the receiver andtransmitter antenna indices, respectively. FIG. 2 illustrates a ULA orfour pairs of ULA elements indexed 1-8. FIG. 3 illustrates four pairs ofcross-polarized arrays. The indexing for the four pairs ofcross-polarized antennas represents grouping two antennas with the samepolarization, which tend to be more correlated. This is analogous to theindexing of 4 pairs of ULA in FIG. 2.

Proposed Codebook Structure

The following notation is used to define the codebooks below:

W: 4Tx feedback precoding matrix.

W₁: first feedback precoding matrix

W₂: second feedback precoding matrix

i₁: PMI index of W₁

i₂: PMI index of W₂

N: maximum number of layers

N_(TXA): number of transmit antennas

I_(k): (k×k)-dimensional identity matrix

Following the guideline of using the same principle for 4Tx enhancementand 8Tx, the block diagonal grid-of-beam (GoB) structure is used. Thisstructure is common between 4Tx and 8Tx.

W₁ and the associated codebook can be written as follows:

$\begin{matrix}{B = \left\lbrack {b_{0}\mspace{14mu} b_{1}\mspace{14mu} \cdots \mspace{14mu} b_{N - 1}} \right\rbrack} & (2) \\{{\lbrack B\rbrack_{{1 + m},{1 + n}} = ^{j\frac{2\pi \; {mn}}{N}}},{m = 0},1,\cdots,{{\frac{N_{TXA}}{2} - {1\mspace{14mu} n}} = 0},1,\cdots,{N - 1}} & (3) \\{X^{(k)} \in \left\{ {{\left\lbrack {b_{{({N_{b}l})}{mod}\mspace{14mu} N}\mspace{14mu} b_{{({{N_{b}k} + 1})}{mod}\mspace{14mu} N}\mspace{14mu} \cdots \mspace{14mu} b_{{({{N_{b}k} + N_{b} - 1})}{mod}\mspace{14mu} N}} \right\rbrack;{k = 0}},1,\cdots,{\frac{N}{N_{b}} - 1}} \right\}} & (4) \\{{W_{1}^{(k)} = \begin{bmatrix}X^{(k)} & 0 \\0 & X^{(k)}\end{bmatrix}},{C_{1} = \left\{ {W_{1}^{(0)},W_{1}^{(1)},W_{1}^{(2)},\cdots,W_{1}^{{({N\text{/}N_{b}})} - 1}} \right\}}} & (5)\end{matrix}$

Here, different W₁ matrices represent a partitioning (without overlap)in terms of beam angles:

-   -   W₁ is a block diagonal matrix of size X where X is a        (N_(TXA)/2)×Nb matrix. Nb denotes the number of adjacent        (N_(TXA)/2)-Tx DFT beams contained in X. Such a design is able        to synthesize N (N_(TXA)/2)-Tx DFT beams within each        polarization group. For a given N, the spatial oversampling        factor is essentially (N/2). The overall (N_(TXA)/2)-Tx DFf beam        collections are captured in the (N_(TXA)/2)×N matrix B.    -   By using co-phasing in W₂ (described below), the composite        precoder W can synthesize up to N DFT beams.    -   For 4Tx, it should be noted that the Release 8 rank-1 codebook        already contains eight 4Tx DFT beams.    -   The set of W₁ matrices represents (N/Nb)-level partitioning        (hence non-overlapping) of the N beam angles (in X, i.e. each        polarization group).    -   This design results in a codebook size of (N/Nb) for W₁.

If some overlap in the set of beam angles is desired between twodifferent W₁ matrices, the above formulation can be slightly modifiedsuch that two consecutive X matrices consists of some overlapping beamangles. Overlapping in beam angles may be beneficial to reduce “edgeeffects”, i.e. to ensure that a common W₁ matrix can be better chosenfor different resource blocks (RBs) within the same precoding sub-bandwhen sub-band precoding or CSI feedback is used. Herein a sub-bandrefers to a set of continuous physical resource blocks (PRB). Withoverlapping,

$\begin{matrix}{X^{(k)} \in \left\{ {{\left\lbrack {b_{{({N_{b}k\text{/}2})}{mod}\mspace{14mu} N}\mspace{14mu} b_{{({{N_{b}k\text{/}2} + 1})}{mod}\mspace{14mu} N}\mspace{14mu} \cdots \mspace{14mu} b_{{({{N_{b}k\text{/}2} + N_{b} - 1})}{mod}\mspace{14mu} N}} \right\rbrack;{k = 0}},1,\cdots,{\frac{2N}{N_{b}} - 1}} \right\}} & (6) \\{{W_{1}^{(k)} = \begin{bmatrix}X^{(k)} & 0 \\0 & X^{(k)}\end{bmatrix}},{C_{1} = \left\{ {W_{1}^{(0)},W_{1}^{(1)},W_{1}^{(2)},\cdots,W_{1}^{{({2N\text{/}N_{b}})} - 1}} \right\}}} & (7)\end{matrix}$

This enhancement targets MU-MIMO improvement and, therefore, is designedfor rank-1 (and at most rank-2). At the same time, the Release 8 4Txcodebook should still be used at least for SU-MIMO. Keeping in mind thatdynamic switching (switching without RRC configuration) between SU-MIMOand MU-MIMO is the baseline assumption for Release 12, an eNodeB shouldbe able to use the Release 8 4Tx and the enhanced componentsinterchangeably (i.e. the switching between the two components should bedynamic). Thanks to the dual-stage feedback structure and in particularthe W=W₁*W₂ structure, this can be realized in a simple and naturalmanner. The enhanced component can be augmented or combined with theRelease 8 codebook as follows:

-   -   the Release 8 4Tx codebook is used as the codebook for W₂ and        associated with W₁=identity matrix;    -   when PMI₁ indicates that W₁=identity matrix is chosen, CB₂ is        chosen as the original Rel-8 codebook;    -   else, when PMI₁ indicates some other W₁, W₁ and CB₂ are chosen        as the enhanced component.

The above switching/augmentation mechanism features the following:

-   -   best possible 4Tx MU-MIMO codebook enhancement opportunity. This        is because the optimization effort for the new components can be        focused on improving MU-MIMO without the need for considering        the SU-MIMO performance (which is covered by the Release 8 4Tx        codebook). Furthermore, the new components can be designed “from        scratch” as the above augmentation mechanism with Release 8 4Tx        codebook can be done without constraining any of the structures        of the new components.    -   maintain best possible performance for 4Tx SU-MIMO without        additional standardization effort. This comes from the use of        Release 8 4Tx codebook. It is noted that the Release 8 4Tx        codebook offers competitive performance in various antenna and        channel setups including dual-polarized arrays as pointed out        partly due to the inherent block diagonal structure in a number        of the precoder matrices/vectors.    -   achieve flexible frequency-selective precoding with physical        uplink shared channel (PUSCH) mode 3-2. If Release 8 codebook is        not re-used and the new codebook is entirely based on the W=W₁W₂        structure, all subband PMI must fall in the same Grid-of-Beam        (GoB) due to the wideband W₁ constraint. This inevitably limits        the precoding gains of mode 3-2 and system performance. On the        contrary, by augmenting the Release 8 codebook, all Release 8        PMI vectors can be used on each subband independently without        any constraint. This is critical to ensure flexible subband        precoding.

The design for W₂ (based on co-phasing and selection) follows thestructure used for 8Tx codebook design. Co-phasing allows some phaseadjustment between the two polarization groups and generation of 4TxDFITx DFT vectors from two block diagonal 2Tx DFITx DFT matrices. The(group) selection operation allows refinement/adjustment of beam anglesacross RBs within the same sub-band thereby maximizingfrequency-selective precoding gain.

The combination of beam selection and co-phasing in W₂—combined withW₁—should result in a unitary precoder W=W₁W₂.

An example of a complete design with non-overlapping block diagonal GoBaugmentation of (N,Nb) codebook is given in the subsequent sections. Itis straightforward to extend the proposed designs to include adjacent W₁matrix overlapping, but is omitted here due to simplicity.

$\begin{matrix}{{B = \left\lbrack {b_{0}\mspace{14mu} b_{1}\mspace{14mu} \cdots \mspace{14mu} b_{N - 1}} \right\rbrack},{\lbrack B\rbrack_{{1 + m},{1 + n}} = ^{j\frac{2\pi \; {mn}}{N}}},{m = 0},{1\mspace{14mu} = 0},1,\cdots,{N - 1}} & (8) \\{X^{(k)} \in \left\{ {{\left\lbrack {b_{{({N_{b}k})}{mod}\mspace{14mu} N}\mspace{14mu} b_{{({{N_{b}k} + 1})}{mod}\mspace{14mu} N}\mspace{14mu} \cdots \mspace{14mu} b_{{({{N_{b}k} + N_{b} - 1})}{mod}\mspace{14mu} N}} \right\rbrack;{k = 0}},{{\ldots \; N\text{/}N_{b}} - 1}} \right\}} & (9)\end{matrix}$

Rank 1

Assuming (N, Nb)=(8,4) as an example.

$\begin{matrix}{{W_{1} \in C_{1}} = \left\{ {I_{4},\begin{bmatrix}X^{(0)} & 0 \\0 & X^{(0)}\end{bmatrix},\begin{bmatrix}X^{(1)} & 0 \\0 & X^{(1)}\end{bmatrix}} \right\}} & (10)\end{matrix}$

→size-3 (Rel-8 codebook augmented with block diagonal GoB).

When W₁=I₄, then W₂εC_(2,R8Tx4r1),

where C_(2,R8Tx4r1) denotes the Release 8 4Tx rank-1 codebook used forW₂.

When

${W_{1} = {\begin{bmatrix}X^{(k)} & 0 \\0 & X^{(k)}\end{bmatrix}\mspace{14mu} \left( {{k = 0},1} \right)}},$

then

${{W_{2} \in {CB}_{2}} = \left\{ {{\frac{1}{\sqrt{2}}\begin{bmatrix}Y \\Y\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y \\{jY}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y \\{- Y}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y \\{- {jY}}\end{bmatrix}}} \right\}},{Y \in \left\{ {\begin{bmatrix}1 \\0 \\0 \\0\end{bmatrix},\begin{bmatrix}0 \\1 \\0 \\0\end{bmatrix},\begin{bmatrix}0 \\0 \\1 \\0\end{bmatrix},\begin{bmatrix}0 \\0 \\0 \\1\end{bmatrix}} \right\}}$

The rank-1 PMI overhead for this example is shown in Table 1.

TABLE 1 W1 overhead W2 overhead (wideband) (subband) Rank-1 log2(3)bits4-bit

Other values of (N, Nb) are not precluded if justified by sufficientsystem performance gain and reasonable feedback overhead.

Note that the block-diagonal enhancement components are sub-matrices ofa subset of the Release 10 8Tx codebook. As such, the 4Tx GoB componentscan be obtained by pruning the 8Tx codebook for 4Tx MIMO feedback. Thisis discussed in more detail below.

N, Nb can also take other values. For example if (N, Nb)=(16, 4).

$\begin{matrix}{{W_{1} \in C_{1}} = \left\{ {I_{4},\begin{bmatrix}X^{(0)} & 0 \\0 & X^{(0)}\end{bmatrix},\begin{bmatrix}X^{(1)} & 0 \\0 & X^{(1)}\end{bmatrix},\begin{bmatrix}X^{(2)} & 0 \\0 & X^{(2)}\end{bmatrix},\begin{bmatrix}X^{(3)} & 0 \\0 & X^{(3)}\end{bmatrix}} \right\}} & (11)\end{matrix}$

→size-5 (Rel-8 codebook augmented with block diagonal GoB).

When W₁=I₄, then W₂εC_(2,R8Tx4r1).

where C_(2,R8Tx4r1) denotes the Release 8 4Tx rank-1 codebook used forW₂.

When

${W_{1} = {\begin{bmatrix}X^{(k)} & 0 \\0 & X^{(k)}\end{bmatrix}\mspace{14mu} \left( {{k = 0},1,2,3} \right)}},$

then

${{W_{2} \in {CB}_{2}} = \left\{ {{\frac{1}{\sqrt{2}}\begin{bmatrix}Y \\Y\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y \\{jY}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y \\{- Y}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y \\{- {jY}}\end{bmatrix}}} \right\}},{Y \in \left\{ {\begin{bmatrix}1 \\0 \\0 \\0\end{bmatrix},\begin{bmatrix}0 \\1 \\0 \\0\end{bmatrix},\begin{bmatrix}0 \\0 \\1 \\0\end{bmatrix},\begin{bmatrix}0 \\0 \\0 \\1\end{bmatrix}} \right\}}$

The rank-1 PMI overhead for this example is shown in Table 2.

TABLE 2 W1 overhead W2 overhead (wideband) (subband) Rank-1 log2(5)-bits4-bit

It is possible in some embodiments that the Release 12 4Tx codebook doesnot comprise the Release 8 codebook. In that case, the identity matrixI₄ is removed from the W₁ codebook C₁.

It is also possible to increase the value of N (e.g. to 32 or 64). Thishowever increases the C₁ codebook size and feedback overhead, andreduces the span of angular spread that Nb adjacent beams can cover.

Rank 2:

Assuming (N,Nb)=(8,4) as an example:

$\begin{matrix}{{W_{1} \in C_{1}} = \left\{ {I_{4},\begin{bmatrix}X^{(0)} & 0 \\0 & X^{(0)}\end{bmatrix},\begin{bmatrix}X^{(1)} & 0 \\0 & X^{(1)}\end{bmatrix}} \right\}} & (12)\end{matrix}$

→size-3 (Rel-8 codebook augmented with block diagonal GoB).

When W₁=I₄, then W₂εC_(2,R8Tx4r2),

where C_(2,R8Tx4r2) denotes the Release 8 4Tx rank-2 codebook used forW₂.

When

${W_{1} = {\begin{bmatrix}X^{(k)} & 0 \\0 & X^{(k)}\end{bmatrix}\mspace{14mu} \left( {{k = 0},1} \right)}},$

then

${{W_{2} \in {CB}_{2}} = \left\{ {{\frac{1}{\sqrt{2}}\begin{bmatrix}Y \\Y\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y \\{jY}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y \\{- Y}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y \\{- {jY}}\end{bmatrix}}} \right\}},{Y \in \left\{ {\begin{bmatrix}1 \\0 \\0 \\0\end{bmatrix},\begin{bmatrix}0 \\1 \\0 \\0\end{bmatrix},\begin{bmatrix}0 \\0 \\1 \\0\end{bmatrix},\begin{bmatrix}0 \\0 \\0 \\1\end{bmatrix}} \right\}}$

One minor shortcoming of this design is that the W₂ overhead is notconstant, but varying dependent on the W₁ matrix. In particular,

if W₁=I, W₂ overhead is 4-bits per subband; and

if W₁ corresponds to the block-diagonal component, W₂ overhead is 3-bitsper subband.

Because W₁ and W₂ are jointly encoded in the PUSCH feedback mode, jointblind decoding of W₁/W₂ is required at the eNodeB, which may increasethe eNodeB implementation complexity.

As one solution, the two columns in a rank-2 precoding matrix W may bechosen from different beams in one grid.

As an example, the following design is possible.

$\begin{matrix}{{W_{1} \in C_{1}} = \left\{ {I_{4},\begin{bmatrix}X^{(0)} & 0 \\0 & X^{(0)}\end{bmatrix},\begin{bmatrix}X^{(1)} & 0 \\0 & X^{(1)}\end{bmatrix}} \right\}} & (13)\end{matrix}$

→size-3 (Rel-8 codebook augmented with block diagonal GoB).

When W₁→I₄:then W₂εC_(2,R8Tx4r2),

where C_(2,R8Tx4r2) denotes the Release 8 4Tx rank-2 codebook used forW₂.

When

${{W_{2} \in {CB}_{2}} = \left\{ {{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\Y_{1} & {- Y_{2}}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\{jY}_{1} & {- {jY}_{2}}\end{bmatrix}}} \right\}},{\left( {Y_{1},Y_{2}} \right) \in \left\{ {\left( {e_{1}^{4},e_{1}^{4}} \right),\left( {e_{2}^{4},e_{2}^{4}} \right),\left( {e_{3}^{4},e_{4}^{4}} \right),\left( {e_{4}^{4},e_{4}^{4}} \right),\left( {e_{1}^{4},e_{2}^{4}} \right),\left( {e_{2}^{4},e_{3}^{4}} \right),\left( {e_{1}^{4},e_{4}^{4}} \right),\left( {e_{2}^{4},e_{4}^{4}} \right)} \right\}}$

then

$\mspace{20mu} {{{W_{2} \in {CB}_{2}} = \left\{ {{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\Y_{1} & {- Y_{2}}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\{j\; Y_{1}} & {{- j}\; Y_{2}}\end{bmatrix}}} \right\}},{\left( {Y_{1},Y_{2}} \right) \in \left\{ {\left( {e_{1}^{4},e_{1}^{4}} \right),\left( {e_{2}^{4},e_{2}^{4}} \right),\left( {e_{3}^{4},e_{3}^{4}} \right),\left( {e_{4}^{4},e_{4}^{4}} \right),\left( {e_{1}^{4},e_{2}^{4}} \right),\left( {e_{2}^{4},e_{3}^{4}} \right),\left( {e_{1}^{4},e_{4}^{4}} \right),\left( {e_{2}^{4},e_{4}^{4}} \right)} \right\}}}$

where e_(q) ^(p) is a p×1 column vector with all elements equivalent tozero, except the q-th element which is 1. The W₂ overhead is 4-bits persubband, consistent with the Release 8 codebook overhead. Note that anyother (Y₁, Y₂) pair, denoted by (e_(m) ⁴,e_(n) ⁴), are equallyapplicable, where 1≦m≦4, 1≦n≦4, m≠n.

Note that in any of the (Y₁, Y₂) pair above, the two selection vectorsinside the brackets [ ] can be permuted. For example, (e₁ ⁴,e₂ ⁴) can bereplaced by (e₂ ⁴,e₁ ⁴), and the resulting codebook is equivalentlyapplicable.

It should be noted that that the block-diagonal enhancement componentsare sub-matrices of a subset of the Release 10 8Tx codebook.

As another solution, it can be resolved by adopting a (N, Nb)=(16, 8)codebook enhancement.

$\begin{matrix}{{W_{1} \in C_{1}} = \; \left\{ {I_{4},\begin{bmatrix}X^{(0)} & 0 \\0 & X^{(0)}\end{bmatrix},\begin{bmatrix}X^{(1)} & 0 \\0 & X^{(1)}\end{bmatrix}} \right\}} & (14)\end{matrix}$

→size-3 (Rel-8 codebook augmented with block diagonal GoB).

When W₁=I₄, then W₂εC_(2,R8Tx4r2),

where C_(2,R8Tx4r2) denotes the Release 8 4Tx rank-2 codebook used forW₂.

When

${W_{1} = {\begin{bmatrix}X^{(k)} & 0 \\0 & X^{(k)}\end{bmatrix}\left( {{k = 0},1} \right)}},$

then

${{W_{2} \in {CB}_{2}} = \left\{ {{\frac{1}{\sqrt{2}}\begin{bmatrix}Y & Y \\Y & {- Y}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y & Y \\{j\; Y} & {{- j}\; Y}\end{bmatrix}}} \right\}},{Y \in \left\{ {\begin{bmatrix}1 \\0 \\0 \\0 \\0 \\0 \\0 \\0\end{bmatrix},\begin{bmatrix}0 \\1 \\0 \\0 \\0 \\0 \\0 \\0\end{bmatrix},\begin{bmatrix}0 \\0 \\1 \\0 \\0 \\0 \\0 \\0\end{bmatrix},\begin{bmatrix}0 \\0 \\0 \\1 \\0 \\0 \\0 \\0\end{bmatrix},\begin{bmatrix}0 \\0 \\0 \\0 \\1 \\0 \\0 \\0\end{bmatrix},\begin{bmatrix}0 \\0 \\0 \\0 \\0 \\1 \\0 \\0\end{bmatrix},\begin{bmatrix}0 \\0 \\0 \\0 \\0 \\0 \\1 \\0\end{bmatrix},\begin{bmatrix}0 \\0 \\0 \\0 \\0 \\0 \\0 \\1\end{bmatrix},} \right\}}$

The rank-2 PMI overhead for this example is shown in Table 3.

TABLE 3 W1 overhead W2 overhead (wideband) (subband) Rank-2 log2(3)-bit4-bit

Alternatively, it is possible that the Release 12 4Tx codebook does notcomprise of the Release 8 codebook. In that case, the identity matrix isremoved from the W₁ codebook C₁.

Another possible design is to use (N,Nb)=(16,4) as

$\begin{matrix}{{W_{1} \in C_{1}} = \left\{ {I_{4},\begin{bmatrix}X^{(0)} & 0 \\0 & X^{(0)}\end{bmatrix},\begin{bmatrix}X^{(1)} & 0 \\0 & X^{(1)}\end{bmatrix},\begin{bmatrix}X^{(2)} & 0 \\0 & X^{(2)}\end{bmatrix},\begin{bmatrix}X^{(3)} & 0 \\0 & X^{(3)}\end{bmatrix}} \right\}} & (5)\end{matrix}$

→size-5 (Rel-8 codebook augmented with block diagonal GoB).

When W₁=I₄, then W²εC_(2,R8Tx4r2),

where C_(2,R8Tx4r2) denotes the Release 8 4Tx rank-2 codebook used forW₂.

When

${W_{1} = {\begin{bmatrix}X^{(k)} & 0 \\0 & X^{(k)}\end{bmatrix}\left( {{k = 0},\ldots,3} \right)}},$

then

$\mspace{20mu} {{{W_{2} \in {CB}_{2}} = \left\{ {{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\Y_{1} & {- Y_{2}}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\{j\; Y_{1}} & {{- j}\; Y_{2}}\end{bmatrix}}} \right\}},{\left( {Y_{1},Y_{2}} \right) \in \left\{ {\left( {e_{1}^{4},e_{1}^{4}} \right),\left( {e_{2}^{4},e_{2}^{4}} \right),\left( {e_{3}^{4},e_{3}^{4}} \right),\left( {e_{4}^{4},e_{4}^{4}} \right),\left( {e_{1}^{4},e_{2}^{4}} \right),\left( {e_{2}^{4},e_{3}^{4}} \right),\left( {e_{1}^{4},e_{4}^{4}} \right),\left( {e_{2}^{4},e_{4}^{4}} \right)} \right\}}}$

It is also possible to increase the value of N (e.g. to 32 or 64). Thehowever increases the C₁ codebook size and feedback overhead, andreduces the span of angular spread that Nb adjacent beams can cover.

Rank 3:

The simplest solution is to reuse the Release 8 codebook as-is forRelease 12.

W ₁ =I ₄  (16)

→size-1 (Rel-8 codebook only).

W ₂ εC _(2,R8Tx4r3),

where C_(2,R8Tx4r3) denotes the Release 8 4Tx rank-3 codebook used forW₂.

If enhancement based on (N,Nb) structure is warranted by sufficientperformance gain, it can be done in a similar fashion as for rank-1 andrank-2. For instance, based on a (N,Nb)=(4,4) design:

$\begin{matrix}{{W_{1} \in C_{1}} = \left\{ {I_{4},\begin{bmatrix}X^{(0)} & 0 \\0 & X^{(0)}\end{bmatrix}} \right\}} & (17)\end{matrix}$

→size-2 (Rel-8 codebook augmented with block diagonal GoB).

When W₁=I₄, then W₂εC_(2,R8Tx4r3),

where C_(2,R8Tx4r3) denotes the Release 8 4Tx rank-3 codebook used forW₂.

When

${W_{1} = {\begin{bmatrix}X^{(k)} & 0 \\0 & X^{(k)}\end{bmatrix}\left( {k = 0} \right)}},$

then

$\mspace{20mu} {{W_{2} \in {CB}_{2}} = \left\{ {\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\Y_{1} & {- Y_{2}}\end{bmatrix}} \right\}}$${\left( {Y_{1},Y_{2}} \right) \in \begin{Bmatrix}{\left( {e_{1}^{4},\left\lbrack {e_{1}^{4},e_{3}^{4}} \right\rbrack} \right),\left( {e_{2}^{4},\left\lbrack {e_{2}^{4},e_{4}^{4}} \right\rbrack} \right),\left( {e_{3}^{4},\left\lbrack {e_{1}^{4},e_{3}^{4}} \right\rbrack} \right),\left( {e_{4}^{4},\left\lbrack {e_{2}^{4},e_{4}^{4}} \right\rbrack} \right),} \\{\left( {\left\lbrack {e_{1}^{4},e_{3}^{4}} \right\rbrack,e_{1}^{4}} \right),\left( {\left\lbrack {e_{2}^{4},e_{4}^{4}} \right\rbrack,e_{2}^{4}} \right),\left( {\left\lbrack {e_{1}^{4},e_{3}^{4}} \right\rbrack,e_{3}^{4}} \right),\left( {\left\lbrack {e_{2}^{4},e_{4}^{4}} \right\rbrack,e_{4}^{4}} \right)}\end{Bmatrix}},$

corresponding to i₂=0, . . . , 7.

To make the W₂ overhead consistent for different W₁ matrices, i₂=8, . .. , 15 shall be reserved for W₂ corresponding to W₁ matrices that areenhancement components (e.g. block diagonal).

Alternatively, when

${W_{1} = {\begin{bmatrix}X^{(k)} & 0 \\0 & X^{(k)}\end{bmatrix}\left( {k = 0} \right)}},$

then

$\mspace{20mu} {{W_{2} \in {CB}_{2}} = \left\{ {{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\Y_{1} & {- Y_{2}}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\{j\; Y_{1}} & {{- j}\; Y_{2}}\end{bmatrix}}} \right\}}$$\left( {Y_{1},Y_{2}} \right) \in {\begin{Bmatrix}{\left( {e_{1}^{4},\left\lbrack {e_{1}^{4},e_{3}^{4}} \right\rbrack} \right),\left( {e_{2}^{4},\left\lbrack {e_{2}^{4},e_{4}^{4}} \right\rbrack} \right),\left( {e_{3}^{4},\left\lbrack {e_{1}^{4},e_{3}^{4}} \right\rbrack} \right),\left( {e_{4}^{4},\left\lbrack {e_{2}^{4},e_{4}^{4}} \right\rbrack} \right),} \\{\left( {\left\lbrack {e_{1}^{4},e_{3}^{4}} \right\rbrack,e_{1}^{4}} \right),\left( {\left\lbrack {e_{2}^{4},e_{4}^{4}} \right\rbrack,e_{2}^{4}} \right),\left( {\left\lbrack {e_{1}^{4},e_{3}^{4}} \right\rbrack,e_{3}^{4}} \right),\left( {\left\lbrack {e_{2}^{4},e_{4}^{4}} \right\rbrack,e_{4}^{4}} \right)}\end{Bmatrix}.}$

In this case W2 overhead is 4-bits per sub-band

The rank-3 PMI overhead for this example is shown in Table 4.

TABLE 4 W2 overhead W1 overhead (wideband) (subband) Rank-3 0-bit (w/oaugmentation) 4-bit 1-bit (w/ augmentation)

It is possible that the Release 12 4Tx codebook does not comprise of theRelease 8 codebook. In that case, the identity matrix is removed fromthe W₁ codebook C₁.

Note that in any of the (Y₁, Y₂) pairs above, the two selection vectorsinside the brackets can be permuted. For example, (e₁ ⁴,[e₁ ⁴,e₃ ⁴]) canbe replaced by (e₁ ⁴[e₃ ⁴,e₁ ⁴]), and the resulting codebook isequivalently applicable.

It is not precluded that the Release 12 4Tx rank-3 codebook isredesigned using the GoB component as proposed above, and not includeany Release 8 4Tx rank-3 precoding matrices.

Another alternative GoB design is (N,Nb)=(8,8), where

$\mspace{20mu} {{W_{1} = {\begin{bmatrix}X^{(k)} & 0 \\0 & X^{(k)}\end{bmatrix}\left( {k = 0} \right)}},{and}}$$\mspace{20mu} {{{W_{2} \in {CB}_{2}} = \left\{ {\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\Y_{1} & {- Y_{2}}\end{bmatrix}} \right\}},{or}}$$\mspace{20mu} {{W_{2} \in {CB}_{2}} = \left\{ {{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\Y_{1} & {- Y_{2}}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\{j\; Y_{1}} & {{- j}\; Y_{2}}\end{bmatrix}}} \right\}}$$\left( {Y_{1},Y_{2}} \right) \in \begin{Bmatrix}{\left( {e_{1}^{4},\left\lbrack {e_{1}^{4},e_{5}^{4}} \right\rbrack} \right),\left( {e_{2}^{4},\left\lbrack {e_{2}^{4},e_{6}^{4}} \right\rbrack} \right),\left( {e_{3}^{4},\left\lbrack {e_{3}^{4},e_{7}^{4}} \right\rbrack} \right),\left( {e_{4}^{4},\left\lbrack {e_{4}^{4},e_{8}^{4}} \right\rbrack} \right),} \\{\left( {e_{5}^{4},\left\lbrack {e_{1}^{4},e_{5}^{4}} \right\rbrack} \right),\left( {e_{6}^{4},\left\lbrack {e_{2}^{4},e_{6}^{4}} \right\rbrack} \right),\left( {e_{7}^{4},\left\lbrack {e_{3}^{4},e_{7}^{4}} \right\rbrack} \right),\left( {e_{3}^{4},\left\lbrack {e_{4}^{4},e_{8}^{4}} \right\rbrack} \right),} \\{\left( {\left\lbrack {e_{1}^{4},e_{5}^{4}} \right\rbrack,e_{1}^{4}} \right),\left( {\left\lbrack {e_{2}^{4},e_{6}^{4}} \right\rbrack,e_{2}^{4}} \right),\left( {\left\lbrack {e_{3}^{4},e_{7}^{4}} \right\rbrack,e_{3}^{4}} \right),\left( {\left\lbrack {e_{4}^{4},e_{8}^{4}} \right\rbrack,e_{4}^{4}} \right),} \\{\left( {\left\lbrack {e_{1}^{4},e_{5}^{4}} \right\rbrack,e_{5}^{4}} \right),\left( {\left\lbrack {e_{2}^{4},e_{6}^{4}} \right\rbrack,e_{6}^{4}} \right),\left( {\left\lbrack {e_{3}^{4},e_{7}^{4}} \right\rbrack,e_{7}^{4}} \right),\left( {\left\lbrack {e_{4}^{4},e_{8}^{4}} \right\rbrack,e_{8}^{4}} \right),}\end{Bmatrix}$

Rank 4:

The simplest solution is to reuse Release 8 codebook as-is for Release12.

W ₁ =I ₄  (18)

→Size-1 (Rel-8 codebook only).

W ₂ εC _(2,R8Tx4r4),

where C_(2,R8Tx4r4) denotes the Release 8 4Tx rank-4 codebook used forW₂.

If enhancement based on (N,Nb) structure is warranted by sufficientperformance gain, it can be done in a similar fashion as for rank-1 andrank-2. For instance, based on a (N,Nb)=(4,4) design,

$\begin{matrix}{{W_{1} \in C_{1}} = \; \left\{ {I_{4},\begin{bmatrix}X^{(0)} & 0 \\0 & X^{(0)}\end{bmatrix}} \right\}} & (19)\end{matrix}$

→Size-2 (Rel-8 codebook augmented with block diagonal GoB)

When W₁=I₄, then W₂εC_(2,R8Tx4r4),

where C_(2,R8Tx4r4) denotes the Release 8 4Tx rank-4 codebook used forW₂.

When

${W_{1} = {\begin{bmatrix}X^{(k)} & 0 \\0 & X^{(k)}\end{bmatrix}\left( {k = 0} \right)}},$

then

${{W_{2} \in {CB}_{2}} = \left\{ {{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\Y_{1} & {- Y_{2}}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\{j\; Y_{1}} & {{- j}\; Y_{2}}\end{bmatrix}}} \right\}},{{{where}\left( {Y_{1},Y_{2}} \right)} \in \begin{Bmatrix}{\left( {\left\lbrack {e_{1}^{4},e_{3}^{4}} \right\rbrack,\left\lbrack {e_{1}^{4},e_{3}^{4}} \right\rbrack} \right),\left( {\left\lbrack {e_{2}^{4},e_{4}^{4}} \right\rbrack,\left\lbrack {e_{2}^{4},e_{4}^{4}} \right\rbrack} \right),} \\{\left( {\left\lbrack {e_{1}^{4},e_{3}^{4}} \right\rbrack,\left\lbrack {e_{2}^{4},e_{4}^{4}} \right\rbrack} \right),\left( {\left\lbrack {e_{2}^{4},e_{4}^{4}} \right\rbrack,\left\lbrack {e_{1}^{4},e_{3}^{4}} \right\rbrack} \right),}\end{Bmatrix}},$

corresponding to i₁=0, . . . , 7.

Note there are a total of 8 W₂ matrices (i₂=0, . . . , 7) for

$W_{1} = {\begin{bmatrix}X^{(k)} & 0 \\0 & X^{(k)}\end{bmatrix}.}$

To make the W₂ overhead consistent across all W₁, W₂ can be reserved fori₂=8, . . . , 15.

Alternatively, one may adopt a (N, Nb)=(8,8) codebook as below:

$\begin{matrix}{{W_{1} \in C_{1}} = \left\{ {I_{4},\begin{bmatrix}X^{(0)} & 0 \\0 & X^{(0)}\end{bmatrix}} \right\}} & (20)\end{matrix}$

→Size-2 (Rel-8 codebook augmented with block diagonal GoB).

When W₁=I₄, then W₂εC_(2,R8Tx4r4),

where C_(2,R8Tx4r4) denotes the Release 8 4Tx rank-4 codebook used forW₂.

When

${W_{1} = {\begin{bmatrix}X^{(k)} & 0 \\0 & X^{(k)}\end{bmatrix}\mspace{14mu} \left( {k = 0} \right)}},$

then

${{W_{2} \in {CB}_{2}} = \left\{ {{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\Y_{1} & {- Y_{2}}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\{j\; Y_{1}} & {{- j}\; Y_{2}}\end{bmatrix}}} \right\}},{{{and}\left( {Y_{1},Y_{2}} \right)} \in {\begin{Bmatrix}{\left( {\left\lbrack {e_{1}^{8},e_{5}^{8}} \right\rbrack,\left\lbrack {e_{1}^{8},e_{5}^{8}} \right\rbrack} \right),\left( {\left\lbrack {e_{2}^{8},e_{6}^{8}} \right\rbrack,\left\lbrack {e_{2}^{8},e_{6}^{8}} \right\rbrack} \right),} \\{\left( {\left\lbrack {e_{3}^{8},e_{7}^{8}} \right\rbrack,\left\lbrack {e_{3}^{8},e_{7}^{8}} \right\rbrack} \right),\left( {\left\lbrack {e_{4}^{8},e_{8}^{8}} \right\rbrack,\left\lbrack {e_{4}^{8},e_{8}^{8}} \right\rbrack} \right),} \\{\left( {\left\lbrack {e_{1}^{8},e_{5}^{8}} \right\rbrack,\left\lbrack {e_{2}^{8},e_{6}^{8}} \right\rbrack} \right),\left( {\left\lbrack {e_{1}^{8},e_{5}^{8}} \right\rbrack,\left\lbrack {e_{3}^{8},e_{7}^{8}} \right\rbrack} \right),} \\{\left( {\left\lbrack {e_{2}^{8},e_{6}^{8}} \right\rbrack,\left\lbrack {e_{4}^{8},e_{8}^{8}} \right\rbrack} \right),\left( {\left\lbrack {e_{3}^{8},e_{7}^{8}} \right\rbrack,\left\lbrack {e_{4}^{8},e_{8}^{8}} \right\rbrack} \right),}\end{Bmatrix}.}}$

Note that in any of the (Y₁,Y₂) pair above, the two selection vectorsinside the brackets [ ] can be permuted.

Also note that any (Y₁, Y₂) pair above can be replaced by a differentpair denoted as ([e_(m) ^(N) ^(b) ,e_(m+N/2) ^(N) ^(b) ],[e_(n) ^(N)^(b) ,e_(n+N/2) ^(N) ^(b) ]), where 1≦m≦N/2, 1≦n≦N/2.

The rank-4 PMI overhead for this example is shown in Table 5.

TABLE 5 W1 overhead (wideband) W2 overhead (sub-band) Rank-4 0-bit (w/oaugmentation) 4-bit 1-bit (w/ augmentation)

It is possible that the Release 12 4Tx codebook does not comprise of theRelease 8 codebook. In that case, the identity matrix is removed fromthe W₁ codebook C₁.

It is not precluded that the Release 12 4Tx rank-4 codebook isredesigned using the GoB component as proposed above, and not includesany Release 8 4Tx rank-4 precoding matrices.

The final 4Tx codebook comprises of rank-r codebook, r=1, 2, 3, 4. Foreach rank r, the corresponding rank-r codebook can be constructed bymethods discussed above regarding rank-1 to rank-4 codebooks. It is notprecluded that codebook is enhanced for certain rank(s), while for theother ranks, Release 8 codebook is reused.

Reformulation of Proposed Rank-1 to Rank-4 Codebooks

Using (N, Nb)=(16,4) as an example, the proposed codebook in thesections above may be re-formulated by the equations shown in the tablesbelow. It should be noted that these tables can be easily extended toother (N, Nb) values.

Rank 1-2

If Release 12 4Tx codebook for rank-1 and rank-2 is redesigned by theGoB framework with adjacent beams overlapping, and does not include theRelease 8 codebook, the 4Tx codebooks can be expressed by equations inTables 6-1 and 6-2.

A first PMI value of n₁ε{0, 1, . . . , ƒ(υ)−1} and a second PMI value ofn₂ε{0, 1, . . . , g(υ)−1} correspond to the codebook indices n₁ and n₂given in Table.6-j, where υ is equal to the associated rank value andwhere j=υ, ƒ(υ)({8,8} and g(υ)=(16,16). Interchangeably, the first andsecond precoding matrix indicators are expressed by i₁ and i₂.

The quantities φ_(n) and ν_(m), are expressed by

φ_(n)=^(ejnm/2)  (21)

ν_(m)=└1;e ^(j2nm/16)┘  (22)

Table 6-1 illustrates a codebook for 1-layer CSI reporting according toone embodiment.

TABLE 6-1 i₂ i₁ 0 1 2 3 4 5 6 7 0-7 W⁽¹⁾ _(2i) ₁ _(,0) W⁽¹⁾ _(2i) ₁_(,1) W⁽¹⁾ _(2i) ₁ _(,2) W⁽¹⁾ _(2i) ₁ _(,3) W⁽¹⁾ _(2i) ₁ _(+1,0) W⁽¹⁾_(2i) ₁ _(+1,1) W⁽¹⁾ _(2i) ₁ _(+1,2) W⁽¹⁾ _(2i) ₁ _(+1,3) i₂ i₁ 8 9 1011 12 13 14 15 0-7 W⁽¹⁾ _(2i) ₁ _(+2,0) W⁽¹⁾ _(2i) ₁ _(+2,1) W⁽¹⁾ _(2i)₁ _(+2,2) W⁽¹⁾ _(2i) ₁ _(+2,3) W⁽¹⁾ _(2i) ₁ _(+3,0) W⁽¹⁾ _(2i) ₁ _(+3,1)W⁽¹⁾ _(2i) ₁ _(+3,2) W⁽¹⁾ _(2i) ₁ _(+3,3) where$W_{m,n}^{(1)} = {\frac{1}{\sqrt{4}}\begin{bmatrix}v_{m} \\{\phi_{n}v_{m}}\end{bmatrix}}$

Table 6-2 illustrates a codebook for 2-layer CSI reporting according toone embodiment.

TABLE 6-2 i₂ i₁ 0 1 2 3 0-7 W⁽²⁾ _(2i) ₁ _(,2i) ₁ _(,0) W⁽²⁾ _(2i) ₁_(,2i) ₁ _(,1) W⁽²⁾ _(2i) ₁ _(+1,2i) ₁ _(+1,0) W⁽²⁾ _(2i) ₁ _(+1,2i) ₁_(+1,1) i₂ i₁ 4 5 6 7 0-7 W⁽²⁾ _(2i) ₁ _(+2,2i) ₁ _(+2,0) W⁽²⁾ _(2i) ₁_(+2,2i) ₁ _(+2,1) W⁽²⁾ _(2i) ₁ _(+3,2i) ₁ _(+3,0) W⁽²⁾ _(2i) ₁ _(+3,2i)₁ _(+3,1) i₂ i₁ 8 9 10 11 0-7 W⁽²⁾ _(2i) ₁ _(,2i) ₁ _(+1,0) W⁽²⁾ _(2i) ₁_(,2i) ₁ _(+1,1) W⁽²⁾ _(2i) ₁ _(+1,2i) ₁ _(+2,0) W⁽²⁾ _(2i) ₁ _(+1,2i) ₁_(+2,1) i₂ i₁ 12 13 14 15 0-7 W⁽²⁾ _(2i) ₁ _(,2i) ₁ _(+3,0) W⁽²⁾ _(2i) ₁_(,2i) ₁ _(+3,1) W⁽²⁾ _(2i) ₁ _(+1,2i) ₁ _(+3,0) W⁽²⁾ _(2i) ₁ _(+1,2i) ₁_(+3,1) where$W_{m,m^{\prime},n}^{{(2)}} = {\frac{1}{\sqrt{8}}\begin{bmatrix}v_{m} & v_{m^{\prime}} \\{\phi_{n}v_{m}} & {{- \phi_{n}}v_{m^{\prime}}}\end{bmatrix}}$

If the Release 12 4Tx codebook for rank-1 and rank-2 is designed byaugmenting the existing Release 8 codebook with GoB components, then theRelease 12 4Tx codebook can be expressed as in Table 6-3 and Table 6-4.

Table 6-3 illustrates a codebook for 1-layer CSI reporting according toone embodiment.

TABLE 6-3 i₁ i₂ 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Use precodingmatrix of index i₂ of Release 8 4Tx rank-1 codebook i₁ i₂ 1-8 0 1 2 3 45 6 7 8 9 10 11 12 13 14 15 Corresponding to i₁ = 0-7 in table 6-1

Table 6-4 illustrates a codebook for 2-layer CSI reporting according toone embodiment.

TABLE 6-4 i₁ i₂ 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Use precodingmatrix of index i₂ of Release 8 4Tx rank-1 codebook i₁ i₂ 1-8 0 1 2 3 45 6 7 8 9 10 11 12 13 14 15 Corresponding to i₁ = 0-7 in table 6-2

Rank 3-4

If the Release 12 4Tx codebook for rank-3 and rank-3 is redesigned bythe GoB framework with adjacent beams overlapping, and not includes theRelease 8 codebook, the 4Tx codebooks can be expressed by equations inTable 6-5 and 6-6.

Table 6-5 illustrates a codebook for 3-layer CSI reporting according toone embodiment.

TABLE 6-5 i₂ i₁ 0 1 2 3 0 W⁽³⁾ _(16i) ₁ _(,16i) ₁ _(,16i) ₁ _(+8,0) W⁽³⁾_(16i) ₁ _(,16i) ₁ _(,16i) ₁ _(+8,1) W⁽³⁾ _(16i) ₁ _(+4,16i) ₁ _(+4,16i)₁ _(+12,0) W⁽³⁾ _(16i) ₁ _(+4,16i) ₁ _(+4,16i) ₁ _(+12,1) i2 i₁ 4 5 6 70 W⁽³⁾ _(16i) ₁ _(+8,16i) ₁ _(,16i) ₁ _(+8,0) W⁽³⁾ _(16i) ₁ _(+8,16i) ₁_(,16i) ₁ _(+8,1) W⁽³⁾ _(16i) ₁ _(+12,16i) ₁ _(+4,16i) ₁ _(+12,0) W⁽³⁾_(16i) ₁ _(+12,16i) ₁ _(+4,16i) ₁ _(+12,1) i₂ i₁ 8 9 10 11 0 W⁽³⁾ _(16i)₁ _(,16i) ₁ _(+8,16i) ₁ _(,0) W⁽³⁾ _(16i) ₁ _(,16i) ₁ _(+8,16i) ₁ _(,1)W⁽³⁾ _(16i) ₁ _(+4,16i) ₁ _(+12,16i) ₁ _(+4,0) W⁽³⁾ _(16i) ₁ _(+4,16i) ₁_(+12,16i) ₁ _(+4,1) i₂ i₁ 12 13 14 15 0 W⁽³⁾ _(16i) ₁ _(,16i) ₁_(+8,16i) ₁ _(+8,0) W⁽³⁾ _(16i) ₁ _(,16i) ₁ _(+8,16i) ₁ _(+8,1) W⁽³⁾_(16i) ₁ _(+4,16i) ₁ _(+12,16i) ₁ _(+12,0) W⁽³⁾ _(16i) ₁ _(+4,16i) ₁_(+12,16i) ₁ _(+12,1) where${W_{m,m^{\prime},m^{''},n}^{(3)} = {\frac{1}{\sqrt{12}}\begin{bmatrix}v_{m} & v_{m^{\prime}} & v_{m^{''}} \\{\phi_{n}v_{m}} & {{- \phi_{n}}v_{m^{\prime}}} & {{- \phi_{n}}v_{m^{''}}}\end{bmatrix}}},{{\overset{\sim}{W}}_{m,m^{\prime},m^{''},n}^{(3)} = {\frac{1}{\sqrt{12}}\begin{bmatrix}v_{m} & v_{m^{\prime}} & v_{m^{''}} \\{\phi_{n}v_{m}} & {\phi_{n}v_{m^{\prime}}} & {{- \phi_{n}}v_{m^{''}}}\end{bmatrix}}}$

Table 6-6 illustrates a codebook for 4-layer CSI reporting according toone embodiment.

TABLE 6-6 i₂ i₁ 0 1 2 3 0 W⁽⁴⁾ _(16i) ₁ _(,16i) ₁ _(+8,16i) ₁ _(,16i) ₁_(+8,0) W⁽⁴⁾ _(16i) ₁ _(,16i) ₁ _(+8,16i) ₁ _(,16i) ₁ _(+8,1) W⁽⁴⁾_(16i) ₁ _(+4,16i) ₁ _(+12,16i) ₁ _(+4,16i) ₁ _(+12,0) W⁽⁴⁾ _(16i) ₁_(+4,16i) ₁ _(+12,16i) ₁ _(+4,16i) ₁ _(+12,1) i₂ i₁ 4 5 6 7 0 W⁽⁴⁾_(16i) ₁ _(,16i) ₁ _(+8,16i) ₁ _(+4,16i) ₁ _(+12,0) W⁽⁴⁾ _(16i) ₁_(,16i) ₁ _(+8,16i) ₁ _(+4,16i) ₁ _(+12,1) W⁽⁴⁾ _(16i) ₁ _(+4,16i) ₁_(+12,16i) ₁ _(,16i) ₁ _(+8,0) W⁽⁴⁾ _(16i) ₁ _(+4,16i) ₁ _(+12,16i) ₁_(,16i) ₁ _(+8,1) where$W_{m,m^{\prime},m^{''},m^{''\prime},n}^{(4)} = {\frac{1}{\sqrt{16}}\begin{bmatrix}v_{m} & v_{m^{\prime}} & v_{m^{''}} & v_{{m^{''}}^{\prime}} \\{\phi_{n}v_{m}} & {\phi_{n}v_{m^{\prime}}} & {{- \phi_{n}}v_{m^{''}}} & {{- \phi_{n}}v_{m^{''\prime}}}\end{bmatrix}}$

If the Release 12 4Tx codebook for rank-3 and rank-3 is designed byaugmenting the existing Release 8 codebook with GoB components, theRelease 12 4Tx codebook can be expressed as in Table 6-7 and Table 6-8.

Table 6-7 illustrates a codebook for 3-layer CSI reporting according toone embodiment.

TABLE 6-7 i₁ i₂ 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Use precodingmatrix of index i₂ of Release 8 4Tx rank-3 codebook i₁ i₂ 1-8 0 1 2 3 45 6 7 8 9 10 11 12 13 14 15 Corresponding to i₁ = 0-7 in Table 6-5.

Table 6-8 illustrates a codebook for 4-layer CSI reporting according toone embodiment.

TABLE 6-8 i₁ i₂ 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Use precodingmatrix of index i₂ of Release 8 4Tx rank-4 codebook i₁ i₂ 1-8 0 1 2 3 45 6 7 8 9 10 11 12 13 14 15 For i₂ = 0-7: corresponding to i₁ = 0-7 inTable 6-6. For i₂ = 8-15: reserved.

Alternative Codebook Designs

For the W₁ codebook C₁ proposed above, the GoB component is expressed inthe form of a block diagonal matrix as

$\begin{matrix}{{W_{1} \in C_{1}} = \left\{ \begin{bmatrix}X^{(k)} & 0 \\0 & X^{(k)}\end{bmatrix} \right\}} & (23)\end{matrix}$

where without overlapping:

$\begin{matrix}{{X^{(k)} \in \begin{Bmatrix}{\begin{bmatrix}b_{{({N_{b}k})}{mod}\; N} & b_{{({{N_{b}k} + 1})}{mod}\; N} & \ldots & b_{{({{N_{b}k} + N_{b} - 1})}{mod}\; N}\end{bmatrix};} \\{{k = 0},1,\ldots \mspace{14mu},{\frac{N}{N_{b}} - 1}}\end{Bmatrix}},} & (24)\end{matrix}$

and with overlapping:

$\begin{matrix}{X^{(k)} \in {\begin{Bmatrix}{\begin{bmatrix}b_{{({N_{b}{k/2}})}{mod}\; N} & b_{{({{N_{b}{k/2}} + 1})}{mod}\; N} & \ldots & b_{{({{N_{b}{k/2}} + N_{b} - 1})}{mod}\; N}\end{bmatrix};} \\{{k = 0},1,\ldots \mspace{14mu},{\frac{2\; N}{N_{b}} - 1}}\end{Bmatrix}.}} & (25)\end{matrix}$

Each x^((k)) represents a group of Nb adjacent beams that model acertain angle of arrival and angular spread.

Alternative designs are possible where the block diagonal sub-matrix ofW₁ (i.e. x^((k))) is replaced by linear or non-linear transformations ofx^((k)), for example:

$\begin{matrix}{{{{W_{1}\left( {n,m,k} \right)} \in C_{1}} = \left\{ {\begin{bmatrix}{f_{n}\left( X^{(k)} \right)} & 0 \\0 & {g_{m}\left( X^{(k)} \right)}\end{bmatrix},{k = 0},\ldots}\mspace{14mu} \right\}},} & (26)\end{matrix}$

where ƒ_(n)( ), g_(m)( ), n=0, . . . , m=0, . . . are linear/non-lineartransformation functions.

In the following section several such possible designs are describedassuming (N,Nb)=(16,4) as an example, but extension to other (N, Nb)values are straightforward.

It is also assumed that the Release 12 4Tx codebook is redesigned usingthe GoB structure and does not include the Release 8 4Tx codebook;however it is straightforward to augment the Release 8 4Tx codebook bythe designs proposed below.

Example 1 Beam Shifting

In one embodiment,

$\begin{matrix}{{{W_{1}\left( {n,m,k} \right)} \in C_{1}} = {\left\{ {\begin{bmatrix}{f_{n}\left( X^{(k)} \right)} & 0 \\0 & {g_{m}\left( X^{(k)} \right)}\end{bmatrix},{k = 0},\ldots}\mspace{14mu} \right\} = {\begin{bmatrix}X^{(k)} & 0 \\0 & {{D(m)}X^{(k)}}\end{bmatrix}.}}} & (27)\end{matrix}$

Alternatively,

$\begin{matrix}{C_{1} = {\begin{bmatrix}{{D(m)}X^{(k)}} & 0 \\0 & X^{(k)}\end{bmatrix}.}} & (28)\end{matrix}$

D(m) (m=0, 1 . . . N−1) is a Nt/2×Nt/2 diagonal matrix, in the case ofNt=4 denoted as

$\begin{matrix}{D = {\begin{bmatrix}1 & 0 \\0 & ^{j\frac{2\pi \; n}{N}}\end{bmatrix}.}} & (29)\end{matrix}$

Herein D(m) performs beam shifting. For instance, as the firstsub-matrix x^((k)) comprises of Nb adjacent beams b_((N) _(b) _(k)mod N)b_((N) _(b) _(k+1)mod N) . . . b_((N) _(b) _(k+N) _(b) _(−1)mod N), thesecond sub-matrix D(m)X^((k)) comprises a different grid of Nb beams as

b _((N) _(b) _(k+m)mod N) b _((N) _(b) _(k+1+m)mod N) . . . b _((N) _(b)_(k+N) _(b) _(−1+m)mod N).

In other words, the second grid of beam is shifted by m beams, where mcan take values from 0 to N−1. If m=0, . . . N−1, the W₁ codebook sizeis increased to

$N\frac{N}{N_{b}}$

without W₁ overlapping, and

$N\frac{2\; N}{N_{b}}$

with W₁ overlapping. Note that the codebook proposed in the initialsections above is a special case where m=0, in which case there is nobeam shifting.

FIG. 4 illustrates an example in which a grid of beams is shifted by m=2beams. A first grid of Nb beams 401 is selected from N beams 402. Thesecond grid of beams 403 is selected from the N beams 402, but isshifted by two beams 404, 405 relative to the first grid of beams 401.

It is not precluded that a subset of D(m) matrices are used inconstructing codebook C₁, where mεΠ, Π⊂ {0, . . . N−1}. For instance,when Π={1}, the Nb beams in the second grid of beams (e.g., for thevertical polarization array 402) are all shifted by 1 beam, which ishalf of the number of overlapping beams (Nb/2=2) between two consecutiveW₁ matrices. As another example, when Π={0,1}, the second grid of Nbbeams may be unshifted, or shifted by one beam.

In another embodiment, both the first and second grid of beams may beshifted. This is expressed as:

$\begin{matrix}{{{{{W_{1}\left( {n,m,k} \right)} \in C_{1}} = {\left\{ {\begin{bmatrix}{f_{n}\left( X^{(k)} \right)} & 0 \\0 & {g_{m}\left( X^{(k)} \right)}\end{bmatrix},{k = 0},\ldots}\mspace{14mu} \right\} = \begin{bmatrix}{{G(n)}X^{(k)}} & 0 \\0 & {{D(m)}X^{(k)}}\end{bmatrix}}},\mspace{20mu} {where}}\mspace{20mu} {{{G(n)} = \begin{bmatrix}1 & 0 \\0 & ^{j\frac{2\pi \; n}{N}}\end{bmatrix}},\left( {{n = 0},{{1\mspace{14mu} \ldots \mspace{14mu} N} - 1}} \right),\mspace{20mu} {{D(m)} = \begin{bmatrix}1 & 0 \\0 & ^{j\frac{2\pi \; n}{N}}\end{bmatrix}},{\left( {{m = 0},{{1\mspace{14mu} \ldots \mspace{14mu} N} - 1}} \right).}}} & (30)\end{matrix}$

Similarly, it is possible to use a subset of G(n) and D(m) matrices ingenerating the codebook C₁.

Example 2 Beam Permutation

It is further possible to permute the selected beams in one or bothsub-matrices of W₁. As one example, the W₁ codebook is expressed as:

$\begin{matrix}{{{{W_{1}\left( {l,m,k} \right)} \in C_{1}} = \left\{ \begin{bmatrix}X^{(k)} & 0 \\0 & {{D(m)}X^{(k)}{P(l)}}\end{bmatrix} \right\}},} & (31)\end{matrix}$

where P(l) is a 4×4 column permutation of an N_(b)×N_(b) identity matrixI_(N) _(b) .

One example of P(l) is

${P(l)} = {\begin{bmatrix}0 & 1 & 0 & 0 \\1 & 0 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{bmatrix}.}$

By multiplying P(l) to D(m)X^((k)), the Nb beams in the secondsub-matrix D(m)X^((k)) are permuted before being co-phased with thefirst grid of beam X^((k)), providing further diversity gain in the W₁codebook.

Permutation can be performed for both sub-matrices of W₁. In anotherexample, the W₁ codebook is expressed a

$\begin{matrix}{{{W_{1}\left( {j,l,n,m,k} \right)} \in C_{1}} = {\left\{ {\begin{bmatrix}{{f_{n}\left( X^{(k)} \right)}{P_{1}(j)}} & 0 \\0 & {{g_{m}\left( X^{(k)} \right)}{P_{2}(l)}}\end{bmatrix},{k = 0},\ldots}\mspace{14mu} \right\} = {\quad\begin{bmatrix}{{D(n)}X^{(k)}{P_{1}(j)}} & 0 \\0 & {{D(m)}X^{(k)}{P_{2}(l)}}\end{bmatrix}}}} & (32)\end{matrix}$

where P₁(j) and P₂(l) perform column permutations for the first and thesecond grid of beams, respectively.

Note that the permutation operation P(l) can be applied without beamshifting (e.g. D(m)).

Example 3 Phase Rotation

In another embodiment,

$\begin{matrix}{{{W_{1}\left( {n,m,k} \right)} \in C_{1}} = {\left\{ {\begin{bmatrix}{f_{n}\left( X^{(k)} \right)} & 0 \\0 & {g_{m}\left( X^{(k)} \right)}\end{bmatrix},{k = 0},\ldots}\mspace{14mu} \right\} = \begin{bmatrix}X^{(k)} & 0 \\0 & {X^{(k)}{D(m)}}\end{bmatrix}}} & (33)\end{matrix}$

Alternatively,

$\begin{matrix}{C_{1} = \begin{bmatrix}{X^{(k)}{D(m)}} & 0 \\0 & X^{(k)}\end{bmatrix}} & (34)\end{matrix}$

D(m) (m=0, 1 . . . N−1) is a N_(b)×N_(b) diagonal matrix denoted as

$\begin{matrix}{D = {\begin{bmatrix}1 & 0 & 0 & 0 \\0 & ^{j\frac{2\; \pi \; n}{N}} & 0 & 0 \\0 & 0 & \ldots & \; \\0 & 0 & 0 & ^{j\frac{2\; {\pi {({N_{k} - 1})}}m}{N}}\end{bmatrix}.}} & (35)\end{matrix}$

Herein D(m) performs phase correction to the Nb beams. For instance, asthe first sub-matrix X^((k)) comprises Nb adjacent beams defined as:

b _((N) _(b) _(k)mod N) b _((N) _(b) _(k+1)mod N) . . . b _((N) _(b)_(k+N) _(b) _(−1)mod N),

the second sub-matrix D(m)X^((k)) comprises a different grid of Nb beamsdefined as

$\begin{matrix}b_{{({{N_{b}k} + m})}{mod}\; N} & {^{j\frac{2\; \pi \; n}{N}}b_{{({{N_{b}k} + 1})}{mod}\; N}} & \ldots & {^{j\frac{2\; {\pi {({N_{b} - 1})}}m}{N}}{b_{{({{N_{b}k} + N_{b} - 1})}{mod}\mspace{11mu} N}.}}\end{matrix}$

In other words, the m-th beam in the second grid (m=0, 1, . . . , Nb−1)is phase rotated by

$\frac{2\; {\pi \left( {N_{k} - 1} \right)}m}{N}$

degrees.

It is also possible to apply the phase rotation to both the first andthe second grid of beams.

An alternative formulation of the phase rotation matrix D(m) is given as

$\begin{matrix}{D = {\begin{bmatrix}1 & 0 & 0 & 0 \\0 & ^{j\frac{\pi}{2N_{b}}} & 0 & 0 \\0 & 0 & \ldots & \; \\0 & 0 & 0 & ^{j\frac{{({N_{k} - 1})}m}{2N_{b}}}\end{bmatrix}.}} & (36)\end{matrix}$

where Nb phase correction components

$\left( {{e.g.},1,^{j\frac{\pi}{2N_{b}}},{\ldots \mspace{14mu} ^{j\frac{{({N_{b} - 1})}\pi}{2N_{b}}}}} \right)$

uniformly sample a 90-degree sector. As such, co-phasing between twogrids of beams is no longer limited to the QPSK alphabet as in the W2codebook, but can take values of

$\left\{ {0,\frac{\pi}{2N_{b}},{\ldots \mspace{14mu} \frac{\left( {N_{b} - 1} \right)\pi}{2N_{b}}}} \right\} + {\left\{ {0,{\pi/2},\pi,,{3{\pi/2}},} \right\}.}$

It will be further understood that combination of beam shifting beampermutation, and/or phase rotation schemes can be used in constructingthe W₁ codebook.

Pruning the 8Tx Codebook for 4Tx MIMO

The LTE Release 10 8Tx codebook is designed using the GoB structure.

Specifically:

each 4Tx polarization array is over-sampled by N DFT beams'

each wideband W₁ matrix comprises Nb adjacent DFT beams to cover acertain AoD and angular spread, and

narrowband W_(Z) performs beam selection and co-phasing.

As such, the 4Tx GoB codebook components can be selected as sub-matricesof a subset of the 8Tx codebook in Release 12. In other words, each 4TxGoB precoder may correspond to four selected rows of a Release 10 8Txprecoder (i.e., pruning the 8Tx codebook down to four rows).

To illustrate this pruning, the Release 10 8Tx codebook is denoted as

C ⁽⁸⁾ =C ₁ ⁽⁸⁾ ×C ₂ ⁽⁸⁾,

where C⁽⁸⁾={W⁽⁸⁾}, C₁ ⁽⁸⁾={W₁ ⁽⁸⁾} and C₂ ⁽⁸⁾={W₂ ⁽⁸⁾} are the first andsecond codebook.

Subsequently, the 4Tx GoB codebook can be written as

C ⁽⁴⁾ =={W ⁽⁴⁾ }=C ₁ ⁽⁴⁾ ×C ₂ ⁽⁴⁾  (37)

where C⁽⁴⁾={W⁽⁴⁾}⊂{W⁽⁸⁾ _([(n1,n2,n3,n4)h])}, and

where, for an 8×R_(matrix) H, H_((nH)) denotes the ni-th row of H,

${H_{\lbrack{{({{n\; 1},\; {n\; 2},\; {n\; 3},\; {n\; 4}})},:}\rbrack} = \begin{pmatrix}H_{({{n\; 1},:})} \\H_{({{n\; 2},:})} \\H_{({{n\; 3},:})} \\H_{({{n\; 4},:})}\end{pmatrix}},$

and (_(n1,n2,n3,n4)) is the row selection vector for codebook pruning.

The 4Tx GoB codebooks proposed above may be pruned from the Release 108Tx codebook. The following notations are used herein:

i₁ ⁴: index of first precoder W₁ for 4Tx

i₂ ⁴ index of second precoder W₂ for 4Tx

i₁ ⁸ index of first precoder W₁ for 8Tx

i₂ ⁸ index of second precoder W₁ for 8Tx

As such, each 8Tx precoding matrix W⁽⁸⁾=W₁ ⁽⁸⁾W₂ ⁽⁸⁾ is denoted by apair of 8Tx codebook indices (i₁ ⁸, i₂ ⁸), and each 4Tx precoding matrixW⁽⁴⁾=W₁ ⁽⁴⁾W₂ ⁽⁴⁾ is denoted by a pair of 4Tx codebook indices (i₁ ⁴,i₂⁴),

Rank-1

For a (N,Nb)=(N,Nb) 4Tx GoB codebook with overlapping beams, whereN<=32, Nb=4, each 4Tx precoding matrix denoted by (i₁ ⁴,i₂ ⁴) can bepruned from a corresponding 8Tx rank-1 precoding matrix denoted by (i₁⁸,i₂ ⁸), where

${l_{1}^{4} = {\frac{32}{N}i_{1}^{8}}},{i_{2}^{4} = i_{2}^{8}},{i_{2}^{4} = 0},\ldots \mspace{14mu},15$

More specifically:

For (N,Nb)=(8,4) 4Tx codebook with overlapping, each 4Tx precodingmatrix denoted by a pair of 4Tx codebook indices (i₁ ⁴,i₂ ⁴) can bepruned from a corresponding 8Tx rank-1 precoding matrix denoted by apair of 8Tx codebook indices (i₁ ⁸,i₂ ⁸), as given in Table 7.

TABLE 7 4Tx indices i₁ ⁴ i₂ ⁴ 0-3 0 1  2  3  4  5  6  7 Corresponding i₁⁸ i₂ ⁸ 8Tx indices 4i₁ ⁴  0 1  2  3  4  5  6  7 4Tx indices i₁ ⁴ i₂ ⁴0-3 8 9 10 11 12 13 14 15 Corresponding i₁ ⁸ i₂ ⁸ 8Tx indices 4i₁ ⁴  8 910 11 12 13 14 15

For a (N,Nb)=(16,4) 4Tx codebook with overlapping beams, each 4Txprecoding matrix denoted by (i₁ ⁴,i₂ ⁴) can be pruned from acorresponding 8Tx rank-1 precoding matrix denoted by (i₁ ⁸,i₂ ⁸), asgiven in Table 8.

TABLE 8 4Tx indices i₁ ⁴ i₂ ⁴ 0-7 0 1  2  3  4  5  6  7 Corresponding i₁⁸ i₂ ⁸ 8Tx indices 2i₁ ⁴  0 1  2  3  4  5  6  7 4Tx indices i₁ ⁴ i₂ ⁴0-7 8 9 10 11 12 13 14 15 Corresponding i₁ ⁸ i₂ ⁸ 8Tx indices 2i₁ ⁴  8 910 11 12 13 14 15

For (N,Nb)=(32,4) 4Tx codebook with overlapping beams, each 4Txprecoding matrix denoted by (i₁ ⁴,i₂ ⁴) can be pruned from acorresponding 8Tx rank-1 precoding matrix denoted by (i₁ ⁸,i₂ ⁸), asgiven in Table 9.

TABLE 9 4Tx indices i₁ ⁴ i₂ ⁴ 0-15 0 1  2  3  4  5  6  7 Correspondingi₁ ⁸ i₂ ⁸ 8Tx indices i₁ ⁴ 0 1  2  3  4  5  6  7 4Tx indices i₁ ⁴ i₂ ⁴0-15 8 9 10 11 12 13 14 15 Corresponding i₁ ⁸ i₂ ⁸ 8Tx indices i₁ ⁴ 8 910 11 12 13 14 15

For a (N,Nb)=(N,Nb) 4Tx codebook without overlapping beams, where N<=32,and Nb=4, each 4Tx precoding matrix denoted by a pair of 4Tx codebookindices (i₁ ⁴,i₂ ⁴) can be pruned from a corresponding 8Tx rank-1precoding matrix denoted by a pair of 8Tx codebook indices (i₁ ⁸,i₂ ⁸),where

${l_{1}^{4} = {\frac{64}{N}i_{1}^{8}}},{i_{2}^{4} = i_{2}^{8}},{i_{2}^{4} = 0},\ldots \mspace{14mu},15.$

More specifically:

For a (N,Nb)=(8,4) 4Tx codebook without overlapping beams, each 4Txprecoding matrix denoted by (i₁ ⁴,i₂ ⁴) can be pruned from acorresponding 8Tx rank-1 precoding matrix denoted by (i₁ ⁸,i₂ ⁸), asgiven in Table 10.

TABLE 10 4Tx indices i₁ ⁴ i₂ ⁴ 0-1 0 1  2  3  4  5  6  7 Correspondingi₁ ⁸ i₂ ⁸ 8Tx indices 8i₁ ⁴  0 1  2  3  4  5  6  7 4Tx indices i₁ ⁴ i₂ ⁴0-1 8 9 10 11 12 13 14 15 Corresponding i₁ ⁸ i₂ ⁸ 8Tx indices 8i₁ ⁴  8 910 11 12 13 14 15

For a (N,Nb)=(16,4) 4Tx codebook without overlapping beams, each 4Txprecoding matrix denoted by (i₁ ⁴,i₂ ⁴) can be pruned from acorresponding 8Tx rank-1 precoding matrix denoted by (i₁ ⁸,i₂ ⁸), asgiven in Table 11.

TABLE 11 4Tx indices i₁ ⁴ i₂ ⁴ 0-3 0 1  2  3  4  5  6  7 Correspondingi₁ ⁸ i₂ ⁸ 8Tx indices 4i₁ ⁴  0 1  2  3  4  5  6  7 4Tx indices i₁ ⁴ i₂ ⁴0-3 8 9 10 11 12 13 14 15 Corresponding i₁ ⁸ i₂ ⁸ 8Tx indices 4i₁ ⁴  8 910 11 12 13 14 15

For a (N,Nb)=(32,4) 4Tx codebook without overlapping beams, each 4Txprecoding matrix denoted by (i₁ ⁴,i₂ ⁴) can be pruned from acorresponding 8Tx rank-1 precoding matrix denoted by (i₁ ⁸,i₂ ⁸), asgiven in Table 12.

TABLE 12 4Tx indices i₁ ⁴ i₂ ⁴ 0-7 0 1  2  3  4  5  6  7 Correspondingi₁ ⁸ i₂ ⁸ 8Tx indices 2i₁ ⁴  0 1  2  3  4  5  6  7 4Tx indices i₁ ⁴ i₂ ⁴0-7 8 9 10 11 12 13 14 15 Corresponding i₁ ⁸ i₂ ⁸ 8Tx indices 2i₁ ⁴  8 910 11 12 13 14 15

Rank-2

Exemplary Codebook 1

In this section, we assume the following 4Tx rank-2 codebook, where theW₂ codebook is size-8.

$\begin{matrix}{{W_{1} = \begin{bmatrix}X^{(k)} & 0 \\0 & X^{(k)}\end{bmatrix}},} & (38)\end{matrix}$

where (k=0, . . . N/N_(b)−1) without overlapping beams, and (k=0, . . .2N/N−1) with overlapping beams.

$\begin{matrix}{{{W_{2} \in {CB}_{2}} = \left\{ {{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\Y_{1} & {- Y_{2}}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\{j\; Y_{1}} & {{- j}\; Y_{2}}\end{bmatrix}}} \right\}},{{{where}\mspace{14mu} \left( {Y_{1},Y_{2}} \right)} \in {\left\{ {\left( {e_{1}^{4},e_{1}^{4}} \right),\left( {e_{2}^{4},e_{2}^{4}} \right),\left( {e_{3}^{4},e_{3}^{4}} \right),\left( {e_{4}^{4},e_{4}^{4}} \right)} \right\}.}}} & (39)\end{matrix}$

For a (N,Nb)=(N,Nb) 4Tx GoB codebook with overlapping beams, whereN<=32, Nb=4, each 4Tx precoding matrix denoted by a pair of 4Tx codebookindices (i₁ ⁴,i₂ ⁴) can be pruned from a corresponding 8Tx rank-2precoding matrix denoted by a pair of 8Tx codebook indices (i₁ ⁸,i₂ ⁸),where

${i_{1}^{4} = {\frac{32}{N}i_{1}^{8}}},{i_{2}^{4} = i_{2}^{8}},{i_{2}^{4} = 0},\ldots \mspace{14mu},7$

More specifically:

For a (N,Nb)=(8,4) 4Tx codebook with overlapping beams, each 4Txprecoding matrix denoted by (i₁ ⁴,i₂ ⁴) can be pruned from acorresponding 8Tx rank-2 precoding matrix denoted by (i₁ ⁸,i₂ ⁸), asgiven in Table 13.

TABLE 13 4Tx indices i₁ ⁴ i₂ ⁴ 0-3 0 1 2 3 4 5 6 7 Corresponding i₁ ⁸ i₂⁸ 8Tx indices 4i₁ ⁴  0 1 2 3 4 5 6 7

For a (N,Nb)=(16,4) 4Tx codebook with overlapping beams, each 4Txprecoding matrix denoted by (i₁ ⁴,i₂ ⁴) can be pruned from acorresponding 8Tx rank-2 precoding matrix denoted by (i₁ ⁸,i₂ ⁸), asgiven in Table 14.

TABLE 14 4Tx indices i₁ ⁴ i₂ ⁴ 0-7 0 1 2 3 4 5 6 7 Corresponding i₁ ⁸ i₂⁸ 8Tx indices 2i₁ ⁴  0 1 2 3 4 5 6 7

For a (N,Nb)=(32,4) 4Tx codebook with overlapping beams, each 4Txprecoding matrix denoted by (i₁ ⁴,i₂ ⁴) can be pruned from acorresponding 8Tx rank-2 precoding matrix denoted by (i₁ ⁸,i₂ ⁸), asgiven in Table 15.

TABLE 15 4Tx indices i₁ ⁴ i₂ ⁴ 0-15 0 1 2 3 4 5 6 7 Corresponding i₁ ⁸i₂ ⁸ 8Tx indices i₁ ⁴ 0 1 2 3 4 5 6 7

For a (N,Nb)=(N,Nb) 4Tx codebook without overlapping beams, where N<=32,Nb=4, each 4Tx precoding matrix denoted by a pair of 4Tx codebookindices (i₁ ⁴,i₂ ⁴) can be pruned from a corresponding 8Tx rank-2precoding matrix denoted by a pair of 8Tx codebook indices (i₁ ⁸,i₂ ⁸),where

${i_{1}^{4} = {\frac{64}{N}i_{1}^{8}}},{i_{2}^{4} = i_{2}^{8}},{i_{2}^{4} = 0},\ldots \;,7$

More specifically:

For a (N,Nb)=(8,4) 4Tx codebook without overlapping beams, each 4Txprecoding matrix denoted by (i₁ ⁴,i₂ ⁴) can be pruned from acorresponding 8Tx rank-2 precoding matrix denoted by (i₁ ⁸,i₂ ⁸), asgiven in Table 16.

TABLE 16 4Tx indices i₁ ⁴ i₂ ⁴ 0-1 0 1 2 3 4 5 6 7 Corresponding i₁ ⁸ i₂⁸ 8Tx indices 8i₁ ⁴  0 1 2 3 4 5 6 7

For a (N,Nb)=(16,4) 4Tx codebook without overlapping beams, each 4Txprecoding matrix denoted by (i₁ ⁴,i₂ ⁴) can be pruned from acorresponding 8Tx rank-2 precoding matrix denoted by (i₁ ⁸,i₂ ⁸), asgiven in Table 17.

TABLE 17 4Tx indices i₁ ⁴ i₂ ⁴ 0-3 0 1 2 3 4 5 6 7 Corresponding i₁ ⁸ i₂⁸ 8Tx indices 4i₁ ⁴  0 1 2 3 4 5 6 7

For a (N,Nb)=(32,4) 4Tx codebook without overlapping beams, each 4Txprecoding matrix denoted by (i₁ ⁴,i₂ ⁴) can be pruned from acorresponding 8Tx rank-2 precoding matrix denoted by (i₁ ⁸,i₂ ⁸), asgiven in Table 18.

TABLE 18 4Tx indices i₁ ⁴ i₂ ⁴ 0-7 0 1 2 3 4 5 6 7 Corresponding i₁ ⁸ i₂⁸ 8Tx indices 2i₁ ⁴  0 1 2 3 4 5 6 7

Exemplary Codebook 2

In this section, we assume the following 4Tx GoB codebook where the W₂codebook is size-16.

$\begin{matrix}{{W_{1} = \begin{bmatrix}X^{(k)} & 0 \\0 & X^{(k)}\end{bmatrix}},} & (40)\end{matrix}$

where (k=0, . . . N/N_(b)−1) without overlapping beams, and (k=0, . . .2N/N_(b)−1) with overlapping beams.

$\begin{matrix}{\mspace{79mu} {{{W_{2} \in {CB}_{2}} = \left\{ {{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\Y_{1} & {- Y_{2}}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\{j\; Y_{1}} & {{- j}\; Y_{2}}\end{bmatrix}}} \right\}},{\left( {Y_{1},Y_{2}} \right) \in {\left\{ {\left( {e_{1}^{4},e_{1}^{4}} \right),\left( {e_{2}^{4},e_{2}^{4}} \right),\left( {e_{3}^{4},e_{3}^{4}} \right),\left( {e_{4}^{4},e_{4}^{4}} \right),\left( {e_{1}^{4},e_{2}^{4}} \right),\left( {e_{2}^{4},e_{3}^{4}} \right),\left( {e_{1}^{4},e_{4}^{4}} \right),\left( {e_{2}^{4},e_{4}^{4}} \right)} \right\}.}}}} & (41)\end{matrix}$

Such a 4Tx codebook cannot be completely pruned out of the 8Tx codebookunless the oversampling rate is N=32, which is equivalent to the 8Txoversampling rate. In this case, when there is W₁ overlapping for 4Tx,each 4Tx precoding matrix denoted by a pair of 4Tx codebook indices (i₁⁴,i₂ ⁴) can be pruned from a corresponding 8Tx rank-2 precoding matrixdenoted by a pair of 8Tx codebook indices (i₁ ⁸,i₂ ⁸), as given in Table19, where

i ₁ ⁴ =i ₁ ⁸ ,i ₁ ⁴=0, . . . 15,

i ₂ ⁴ =i ₂ ⁸ ,i ₂ ⁴=0, . . . ,15

TABLE 19 4Tx indices i₁ ⁴ i₂ ⁴ 0-15 0 1  2  3  4  5  6  7 Correspondingi₁ ⁸ i₂ ⁸ 8Tx indices i₁ ⁴ 0 1  2  3  4  5  6  7 4Tx indices i₁ ⁴ i₂ ⁴0-15 8 9 10 11 12 13 14 15 Corresponding i₁ ⁸ i₂ ⁸ 8Tx indices i₁ ⁴ 8 910 11 12 13 14 15

For a (N,Nb)=(32,4) 4Tx GoB codebook without overlapping beams, each 4Txprecoding matrix denoted by a pair of 4Tx codebook indices (i₁ ⁴,i₂ ⁴)can be pruned from a corresponding 8Tx rank-2 precoding matrix denotedby a pair of 8Tx codebook indices (i₁ ⁸,i₂ ⁸), as given in Table 20,where

i ₁ ⁴=2i ₁ ⁸ ,i ₁ ⁴=0, . . . ,7,

i ₂ ⁴ =i ₂ ⁸ ,i ₂ ⁴=0, . . . ,15

TABLE 20 4Tx indices i₁ ⁴ i₂ ⁴ 0-7 0 1  2  3  4  5  6  7 Correspondingi₁ ⁸ i₂ ⁸ 8Tx indices 2i₁ ⁴  0 1  2  3  4  5  6  7 4Tx indices i₁ ⁴ i₂ ⁴0-7 8 9 10 11 12 13 14 15 Corresponding i₁ ⁸ i₂ ⁸ 8Tx indices 2i₁ ⁴  8 910 11 12 13 14 15

Rank-3

A rank-3 4Tx GoB codebook cannot be pruned from the rank-3 8Tx codebook.This can be seen from the 8Tx design, which followed the (N,Nb)=(16,8)structure, where

$\begin{matrix}{{{B = \left\lbrack {b_{0}\mspace{14mu} b_{1}\mspace{14mu} \ldots \mspace{14mu} b_{15}} \right\rbrack},{\lbrack B\rbrack_{{1 + m},{1 + n}} = ^{j\frac{2}{16}}},{m = 0},1,2,3,{n = 0},1,\ldots \;,15}{X^{(k)} \in \begin{Bmatrix}\begin{bmatrix}{b_{4k\; {mod}\; 32}\mspace{11mu} b_{({{4k} + 1})}\mspace{14mu} b_{{({{4k} + 2})}{mod}\; 32}\mspace{14mu} b_{{({{4k} + 3})}{mod}\; 32}\mspace{14mu} b_{{({{4k} + 4})}\; {mod}\; 32}} \\{b_{{{{{*4k} + 5})}{mod}\; 32}\mspace{14mu}}b_{{({{4k} + 6})}\; {mod}\; 32}\mspace{14mu} b_{{({{4k} + 7})}\; {mod}\; 32}}\end{bmatrix} \\{{k = 0},{\ldots \; 3}}\end{Bmatrix}}{{W_{1}^{(k)} = \begin{bmatrix}X^{(k)} & 0 \\0 & X^{(k)}\end{bmatrix}},{{{Codebook}\; 1\text{:}\mspace{14mu} C_{1}} = \left\{ {W_{1}^{(0)},W_{1}^{(1)},W_{1}^{(2)},W_{1}^{(3)}} \right\}}}} & (42) \\{\mspace{79mu} {{{W_{2} \in {CB}_{2}} = \left\{ {\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\Y_{1} & {- Y_{2}}\end{bmatrix}} \right\}},}} & (43) \\{\left( {Y_{1},Y_{2}} \right) \in {\begin{Bmatrix}{\left( {e_{1}^{4},\left\lbrack {e_{1}^{4},e_{5}^{4}} \right\rbrack} \right),\left( {e_{2}^{4},\left\lbrack {e_{2}^{4},e_{6}^{4}} \right\rbrack} \right),\left( {e_{3}^{4},\left\lbrack {e_{3}^{4},e_{7}^{4}} \right\rbrack} \right),\left( {e_{4}^{4},\left\lbrack {e_{4}^{4},e_{8}^{4}} \right\rbrack} \right),} \\{\left( {e_{5}^{4},\left\lbrack {e_{1}^{4},e_{5}^{4}} \right\rbrack} \right),\left( {e_{6}^{4},\left\lbrack {e_{2}^{4},e_{6}^{4}} \right\rbrack} \right),\left( {e_{7}^{4},\left\lbrack {e_{3}^{4},e_{7}^{4}} \right\rbrack} \right),\left( {e_{3}^{4},\left\lbrack {e_{4}^{4},e_{8}^{4}} \right\rbrack} \right),} \\{\left( {\left\lbrack {e_{1}^{4},e_{5}^{4}} \right\rbrack,e_{1}^{4}} \right),\left( {\left\lbrack {e_{2}^{4},e_{6}^{4}} \right\rbrack,e_{2}^{4}} \right),\left( {\left\lbrack {e_{3}^{4},e_{7}^{4}} \right\rbrack,e_{3}^{4}} \right),\left( {\left\lbrack {e_{4}^{4},e_{8}^{4}} \right\rbrack,e_{4}^{4}} \right),} \\{\left( {\left\lbrack {e_{1}^{4},e_{5}^{4}} \right\rbrack,e_{5}^{4}} \right),\left( {\left\lbrack {e_{2}^{4},e_{6}^{4}} \right\rbrack,e_{6}^{4}} \right),\left( {\left\lbrack {e_{3}^{4},e_{7}^{4}} \right\rbrack,e_{7}^{4}} \right),\left( {\left\lbrack {e_{4}^{4},e_{8}^{4}} \right\rbrack,e_{8}^{4}} \right),}\end{Bmatrix}.}} & (44)\end{matrix}$

The column vectors of each rank-3 8Tx precoding matrix comprisecritically sampled (e.g. sampling rate 4) 4Tx DFT vectors in order toensure that a rank-3 8Tx precoding matrix satisfies the unitaryconstraint. For 4Tx, in order to achieve the unitary constraint, columnvectors of each rank-3 4Tx precoding matrix must comprise criticallysampled (e.g. sampling rate 2) 2Tx DFT vectors. As such, the columnvectors of any 4Tx GoB matrix will always span across different W₁matrices of the 8Tx codebook, leaving it impossible to construct therank-3 4-Tx codebook by pruning the 8Tx codebook.

However the rank-3 4Tx GoB component can be pruned from the rank-6 8Txcodebook, where

C ⁽⁴⁾ ={W ⁽⁴⁾}_(rank3) ⊂{W ⁽⁸⁾_([(n1,n2,n3,n4),(m1,m2,m3)])}_(rank6)  (45)

e.g., the (_(n1,n2,n3,n4))-th row and _((m1,m2,m3))-th column of arank-6 8Tx precoding matrix corresponding to a pair of codebook index(i₁ ⁸,i₂ ⁸) are used to construct a 4Tx precoding matrix correspondingto a pair of codebook indices (i₁ ⁴,i₂ ⁴).

For instance, for the (N,Nb)=(4,4) GoB codebook, the pruning shown inTable 21 is possible:

TABLE 21 i₁ ⁴ i₂ ⁴ 4Tx indices 0 arbitrary i₁ ⁸ i₂ ⁸ Corresponding 8Tx 00 indices

Note that the column-selection method _((m1,m2,m3)) may depend on the4Tx GoB precoding matrix. For instance:

(Y ₁ ,Y ₂)ε{(e ₁ ⁴ ,[e ₁ ⁴ ,e ₃ ⁴])}

(m1,m2,m3)=(1,2,6)  (46)

(Y ₁ ,Y ₂)ε{(e ₃ ⁴ ,[e ₁ ⁴ ,e ₃ ⁴])}

(m1,m2,m3)=(5,2,6)  (47)

(Y ₁ ,Y ₂)ε{([e ₁ ⁴ ,e ₃ ⁴ ],e ₁ ⁴)}

(m1,m2,m3)=(1,5,2)  (48)

(Y ₁ ,Y ₂)ε{([e ₁ ⁴ ,e ₃ ⁴ ],e ₃ ⁴)}

(m1,m2,m3)=(1,5,6)  (49)

Rank-4

Similarly, the rank-4 4Tx GoB component cannot be pruned from the rank-48Tx codebook. However, a rank-4 4Tx GoB component can be pruned from therank-8 8Tx codebook, where

C ⁽⁴⁾ ={W ⁽⁴⁾}_(rank-4) ⊂{W ⁽⁸⁾_([(n1,n2,n3,n4),(m1,m2,m3,m4)])}_(rank-8)  (50)

e.g., the (_(n1,n2,n3,n4))-th row and _((m1,m2,m3,m4))-th column of arank-8 8Tx precoding matrix corresponding to a pair of codebook index(i₁ ⁸,i₂ ⁸) is used to construct a 4Tx precoding matrix corresponding toa pair of codebook indices (i₁ ⁴,i₂ ⁴). For instance for the(N,Nb)=(4,4) GoB codebook, the pruning shown in Table 22 is possible.

TABLE 22 i₁ ⁴ i₂ ⁴ 4Tx indices 0 arbitrary i₁ ⁸ i₂ ⁸ Corresponding 8Tx 00 indices

Note that the column-selection method _((m1,m2,m3,m4)) may depend on the4Tx GoB precoding matrix. For instance:

(Y ₁ ,Y ₂)ε{([e ₁ ⁴ ,e ₃ ⁴ ],[e ₁ ⁴ ,e ₃ ⁴])}

(m1,m2,m3,m4)=(1,5,2,6)  (51)

(Y ₁ ,Y ₂)ε{([e ₂ ⁴ ,e ₄ ⁴ ],[e ₂ ⁴ ,e ₄ ⁴])}

(m1,m2,m3,m4)=(3,7,4,8)  (52)

(Y ₁ ,Y ₂)ε{([e ₁ ⁴ ,e ₃ ⁴ ],[e ₂ ⁴ ,e ₄ ⁴])}

(m1,m2,m3,m4)=(1,5,4,8)  (53)

(Y ₁ ,Y ₂)ε{([e ₂ ⁴ ,e ₄ ⁴ ],[e ₁ ⁴ ,e ₃ ⁴])}

(m1,m2,m3,m4)=(3,7,2,6)  (54)

Row Selection Vector (n1,n2,n3,n4) for Pruning

In one embodiment, the row selection vector (_(n1,n2,n3,n4)) ishard-coded/fixed in the specification, and is not signalled to the UE.FIG. 5 illustrates an example of using a row selection vector to pruneantenna pairs in an 8Tx array. Eight antennas 1-8 are arranged in fourcross-polarized pairs 501-504. Using the row selection vector(_(n1,n2,n3,n4))_(=[1,2,5,6]) prunes the first two cross-polarizedantenna pairs 501, 502 from the 8Tx array 505 for a 4Tx deployment 506.

In another embodiment, the row selection vector (_(n1,n2,n3,n4)) may besemi-statically radio resource control (RRC)-configured for a UE, andcan be different for different UEs. There are multiple possible methodsfor RRC signalling (_(n1,n2,n3,n4))

Example 1

For the k-th antenna port of a 4-Tx MIMO system (k=1, 2, 3, 4), thecNodeB signals the corresponding virtual antenna port index n_(k) in an8-Tx MIMO system. This requires a signaling overhead of log 2(8)×4=12bits.

Example 2

As there are a total of C₄ ⁸=70 ways of choosing 4-antenna ports out ofan 8-antenna port system, log 2(70)=7-bits are used to signal thecombinatorial index of the antenna selection vector (_(n1,n2,n3,n4)).This achieves 5-bits of overhead savings.

It is further possible to down-size the set of candidate antennaselection vectors {_((n1,n2,n3,n4))}, which may further reduce the RRCsignalling overhead. For instance, the eNodeB may use 1-bit to configurethe UE to assume (_(n1,n2,n3,n4))_(=[1,2,5,6]) or(_(n1,n2,n3,n4))_(=[1,4,5,8]). With (_(n1,n2,n3,n4))_(=[1,4,5,6]), twoadjacent cross-polarized antenna pairs are configured, which models 4Txcross-polarized antennas with small antenna spacing. With(_(n1,n2,n3,n4))_(=[1,4,5,8]), two antenna pairs with large antennaspacing are modelled. This may be important for some wireless operatorsthat must deploy wide antenna spacing due to their GSM/HSPA/LTE spectrumfarming limitation, where two separate antenna radomes with largespacing are used for 4Tx MIMO.

For a UE configured in transmission mode 10, where the UE is configuredwith multiple CSI-RS resources, the row selection vector(_(n1,n2,n3,n4)) can be independently configured for each 4-Tx CSI-RSresource.

Example of 4Tx Codebook from 8Tx Pruning

Using the methods proposed above for rank-1,2,3,4Tx GoB codebooks, oneexample of a 4Tx codebook is summarized below.

Release 12 enhancement is achieved by augmenting the Release 8 codebookwith double-codebook (DCB) components.

-   -   component 1 (Release 8): W1=identity matrix, W2 selected from        Release 8 codebook,    -   component 2 (DCB): . . . pruned Release 10 8Tx codebook, select        four rows out of eight rows from the 8Tx precoding matrices        (i.e., a subset of the 8Tx matrices).

Rank-1:

Component 1: Release 8

W ₁ =I ₄,  (55)

W ₂ εC _(2,R8-4Tx-R1)  (56)

Component 2: DCB:

W₁: i₁=0, . . . 15, each W₁ is the (1,2,5,6)^(th) row of the (i₁)^(th)8Tx W₁ codebook.

W₂: i₂=0, . . . 15, same as 8Tx W₂ codebook

-   -   Note: the 4Tx DCB components are the (1,2,5,6)^(th) row of the        8Tx DCB codebook. This is equivalent to a (N,Nb)=(32,4) codebook        with adjacent W₁ overlapping.

Rank-2:

Component 1: Release 8

W ₁ =I ₄,  (57)

W ₂ εC _(2,R8-4Tx-R2)  (58)

Component 2: DCB:

W₁: i₁=0, . . . 15, each W₁ is the (1,2,5,6)^(th) row of the (i₁)^(th)8Tx W₁ codebook.

W₂: i₂=0, . . . 15, same as 8Tx W₂ codebook.

-   -   Note: the 4Tx DCB components are the (1,2,5,6)^(th) row of the        8Tx DCB codebook. This is equivalent to a (N,Nb)=(32,4) codebook        with adjacent W₁ overlapping.

Rank-3/4:

Re-use the Release 8 codebook.

Further Enhancement of 4Tx Codebook

Any of the codebooks proposed above can be further enhanced by addingmore 4T precoding matrices. Note that in the sections above, the Release8 4Tx codebook is inherited in Release 12 4Tx codebook in the form ofW=W₁W₂, where W₁ is a 4×4 identity matrix and W₂ are taken from Release8 codebook. An extension of this design is possible, where the W₁codebook not only includes the 4×4 identity matrix I₄, but also includesother size 4×4 matrices. One exemplary extension is where the W₁codebook comprises of a set of diagonal matrices, where each diagonalelement performs phase rotation for each row of W₂ matrix.

Following this design principle, an example of a further-enhanced 4Txcodebook is given below.

Rank-1/2:

C ₁ ={W ₁ ⁽⁰⁾ ,W ₁ ⁽¹⁾ ,W ₁ ⁽²⁾ , . . . ,W ₁ ^(N-1)},  (59)

where:

For i₁=0, . . . , N/2-1, W₁ ⁽⁰⁾,W₁ ⁽¹⁾,W₁ ⁽²⁾, . . . ,W₁ ^((N/2-1)) arediagonal matrices (e.g. performing phase rotation), and W₂ inherits theRelease 8 4Tx codebook. In particular, for i₁, =0, W₁ ⁽⁰⁾=I₄. Thisallows the Release 8 codebook to be re-used without change in Release12.

When i₁=(N/2), . . . , N−1, then W₁ ^((N/2)) . . . W₁ ^((N-1)) are blockdiagonal matrices designed with double-codebook structure. In this case,W₁ and W₂ can use any of the DCB codebook components proposed in thesections above. As one example, W₁/W₂ may be pruned 8Tx codebooks, whereN=32, and

W₁: i₁−16, . . . 31, each W₁ is the (1,2,5,6)^(th) row of the(i₁-15)^(th) 8Tx W₁ matrix,

W₂: i₂=0, . . . 15, same as 8Tx W₂ codebook.

Herein, the W₁ overhead is 5-bit, and W₂ overhead is 4-bits per subband.

Rank-3 and rank-4 may follow the same enhancement design if necessary.

Alternative Design for Rank-2

The GoB designs proposed above assumed that each W₁ matrix comprises agroup of adjacent DFT beams. It is noted that the beams in each W₁ gridare not necessarily adjacent, which makes other designs possible. Inthis section we propose an example rank-2 design with non-adjacent beamsin W₁.

To recapture the notations, note that

$\begin{matrix}{W_{1}^{(k)} = \begin{bmatrix}X^{(k)} & 0 \\0 & X^{(k)}\end{bmatrix}} & (60)\end{matrix}$

where X^((k)) comprises of multiple 2×1 DFT beams, k=i₁. In thefollowing section, we assume an over-sampling ratio N=16, but theproposed design can be easily generalized to other N values.

Alternative Design 1

It is noted that an N-times oversampled 2×1 DFT beam expressed as

${b_{i\; {mod}\; N} = \begin{bmatrix}1 \\^{j\; 2{d/N}}\end{bmatrix}},{l = 0},{{\ldots \; N} - 1}$

is orthogonal to

b _((1+N/2)mod N).

Therefore each W₁ grid may comprise two orthogonal DFT beams expressedas

$\begin{matrix}{{W_{1}^{(k)} = \begin{bmatrix}X^{(k)} & 0 \\0 & X^{(k)}\end{bmatrix}},} & (61) \\{{X^{(k)} = \left\lbrack {b_{l\; {mod}\; N},b_{{({l + {N/2}})}{mod}\; N}} \right\rbrack},{k = 1.}} & (62)\end{matrix}$

The W₂ codebook may comprise beam selection and co-phasing matrixes, forexample, denoted as:

$\begin{matrix}{W_{2} = {C_{2} = {\Omega \circ \left\{ {\begin{pmatrix}e_{1} & e_{2} \\e_{1} & e_{2}\end{pmatrix},\begin{pmatrix}e_{1} & e_{2} \\e_{1} & {- e_{2}}\end{pmatrix},\begin{pmatrix}e_{1} & e_{2} \\{- e_{1}} & e_{2}\end{pmatrix},\begin{pmatrix}e_{1} & e_{2} \\{- e_{1}} & {- e_{2}}\end{pmatrix}} \right\}}}} & (63)\end{matrix}$

where ∘ denotes the Schur product, e_(i) ² is a 2×1 column vector withall zero entries except the i-th element being 1, an

$\begin{matrix}{\Omega = \left\{ \begin{pmatrix}I_{2} & \; \\\; & {^{j\; 2\pi \; {k/N}}I_{2}}\end{pmatrix} \right\}_{k = \Xi}} & (64)\end{matrix}$

are diagonal matrices that perform co-phasing.

As an example,

$\begin{matrix}{{\Omega = \left\{ {\begin{pmatrix}I_{2} & \; \\\; & I_{2}\end{pmatrix},\begin{pmatrix}I_{2} & \; \\\; & {- \; I_{2}}\end{pmatrix},\begin{pmatrix}I_{2} & \; \\\; & {j\; I_{2}}\end{pmatrix},\begin{pmatrix}I_{2} & \; \\\; & {{- j}\; I_{2}}\end{pmatrix},} \right\}},} & (65)\end{matrix}$

which results in a 4-bits W₂ codebook. Correspondingly, for eachX^((k))=[b_(l mod N),b_((l+N/2)mod N]), k=l=i₁, the correspondingsixteen composite matrices W=W_(i) ₁ _(i) ₂ , i₂=0, . . . 15 areexpressed as shown in Table 23.

TABLE 23 i₂ = 0, . . . , 3 $W_{k,0} = \begin{pmatrix}b_{lmodN} & b_{{({l + {N/2}})}{modN}} \\b_{lmodN} & b_{{({l + {N/2}})}{modN}}\end{pmatrix}$ $W_{k,1} = \begin{pmatrix}b_{lmodN} & b_{{({l + {N/2}})}{modN}} \\b_{lmodN} & {- b_{{({l + {N/2}})}{modN}}}\end{pmatrix}$ $W_{k,2} = \begin{pmatrix}b_{lmodN} & b_{{({l + {N/2}})}{modN}} \\{- b_{lmodN}} & b_{{({l + {N/2}})}{modN}}\end{pmatrix}$ $W_{k,3} = \begin{pmatrix}b_{lmodN} & b_{{({l + {N/2}})}{modN}} \\{- b_{lmodN}} & {- b_{{({l + {N/2}})}{modN}}}\end{pmatrix}$ i₂ = 4, . . . , 7 $W_{k,4} = \begin{pmatrix}b_{lmodN} & b_{{({l + {N/2}})}{modN}} \\{- b_{lmodN}} & {- b_{{({l + {N/2}})}{modN}}}\end{pmatrix}$ $W_{k,5} = \begin{pmatrix}b_{lmodN} & b_{{({l + {N/2}})}{modN}} \\{- b_{lmodN}} & b_{{({l + {N/2}})}{modN}}\end{pmatrix}$ $W_{k,6} = \begin{pmatrix}b_{lmodN} & b_{{({l + {N/2}})}{modN}} \\b_{lmodN} & {- b_{{({l + {N/2}})}{modN}}}\end{pmatrix}$ $W_{k,7} = \begin{pmatrix}b_{lmodN} & b_{{({l + {N/2}})}{modN}} \\b_{lmodN} & b_{{({l + {N/2}})}{modN}}\end{pmatrix}$ i₂ = 8, . . . , 11 $W_{k,8} = \begin{pmatrix}b_{lmodN} & b_{{({l + {N/2}})}{modN}} \\{jb}_{lmodN} & {jb}_{{({l + {N/2}})}{modN}}\end{pmatrix}$ $W_{k,9} = \begin{pmatrix}b_{lmodN} & b_{{({l + {N/2}})}{modN}} \\{jb}_{lmodN} & {- {jb}_{{({l + {N/2}})}{modN}}}\end{pmatrix}$ $W_{k,10} = \begin{pmatrix}b_{lmodN} & b_{{({l + {N/2}})}{modN}} \\{- {jb}_{lmodN}} & {jb}_{{({l + {N/2}})}{modN}}\end{pmatrix}$ $W_{k,11} = \begin{pmatrix}b_{lmodN} & b_{{({l + {N/2}})}{modN}} \\{- {jb}_{lmodN}} & {- {jb}_{{({l + {N/2}})}{modN}}}\end{pmatrix}$ i₂ = 12, . . . , 15 $W_{k,12} = \begin{pmatrix}b_{lmodN} & b_{{({l + {N/2}})}{modN}} \\{- {jb}_{lmodN}} & {- {jb}_{{({l + {N/2}})}{modN}}}\end{pmatrix}$ $W_{k,13} = \begin{pmatrix}b_{lmodN} & b_{{({l + {N/2}})}{modN}} \\{- {jb}_{lmodN}} & {jb}_{{({l + {N/2}})}{modN}}\end{pmatrix}$ $W_{k,14} = \begin{pmatrix}b_{lmodN} & b_{{({l + {N/2}})}{modN}} \\{jb}_{lmodN} & {- {jb}_{{({l + {N/2}})}{modN}}}\end{pmatrix}$ $W_{k,15} = \begin{pmatrix}b_{lmodN} & b_{{({l + {N/2}})}{modN}} \\{jb}_{lmodN} & {jb}_{{({l + {N/2}})}{modN}}\end{pmatrix}$

It is noted that W_(i) ₁ _(i) ₂ , i₂=4, . . . , 7 are identical to W_(i)₁ _(i) ₂ , i₂=0, . . . , 3, and W_(i) ₁ _(i) ₂ , i₂=12 . . . , 15 areidentical to W_(i) ₁ _(i) ₂ , i₂=8, . . . , 11. Therefore it's possibleto reduce the W₂ size to 3-bits (i₂=0, . . . , 7), where

$\begin{matrix}{{\Omega = \left\{ {\begin{pmatrix}I_{2} & \; \\\; & I_{2}\end{pmatrix},\begin{pmatrix}I_{2} & \; \\\; & {j\; I_{2}}\end{pmatrix}} \right\}},} & (66)\end{matrix}$

and the composite precoder W=W_(i) ₁ _(i) ₂ is given by the values shownin Table 24.

TABLE 24 i₂ = 0, . . . , 3 $W_{k,0} = \begin{pmatrix}b_{lmodN} & b_{{({l + {N/2}})}{modN}} \\b_{lmodN} & b_{{({l + {N/2}})}{modN}}\end{pmatrix}$ $W_{k,1} = \begin{pmatrix}b_{lmodN} & b_{{({l + {N/2}})}{modN}} \\b_{lmodN} & {- b_{{({l + {N/2}})}{modN}}}\end{pmatrix}$ $W_{k,2} = \begin{pmatrix}b_{lmodN} & b_{{({l + {N/2}})}{modN}} \\{- b_{lmodN}} & b_{{({l + {N/2}})}{modN}}\end{pmatrix}$ $W_{k,3} = \begin{pmatrix}b_{lmodN} & b_{{({l + {N/2}})}{modN}} \\{- b_{lmodN}} & {- b_{{({l + {N/2}})}{modN}}}\end{pmatrix}$ i₂ = 4, . . . , 7 $W_{k,4} = \begin{pmatrix}b_{lmodN} & b_{{({l + {N/2}})}{modN}} \\{jb}_{lmodN} & {jb}_{{({l + {N/2}})}{modN}}\end{pmatrix}$ $W_{k,5} = \begin{pmatrix}b_{lmodN} & b_{{({l + {N/2}})}{modN}} \\{jb}_{lmodN} & {- {jb}_{{({l + {N/2}})}{modN}}}\end{pmatrix}$ $W_{k,6} = \begin{pmatrix}b_{lmodN} & b_{{({l + {N/2}})}{modN}} \\{- {jb}_{lmodN}} & {jb}_{{({l + {N/2}})}{modN}}\end{pmatrix}$ $W_{k,7} = \begin{pmatrix}b_{lmodN} & b_{{({l + {N/2}})}{modN}} \\{- {jb}_{lmodN}} & {- {jb}_{{({l + {N/2}})}{modN}}}\end{pmatrix}$

Note that other co-phasing methods Ω are possible, for example

$\begin{matrix}{\Omega = {\left\{ {\begin{pmatrix}I_{2} & \; \\\; & I_{2}\end{pmatrix},\begin{pmatrix}I_{2} & \; \\\; & {j\; I_{2}}\end{pmatrix},\begin{pmatrix}I_{2} & \; \\\; & {^{j\frac{\pi}{4}}\; I_{2}}\end{pmatrix},\begin{pmatrix}I_{2} & \; \\\; & {^{j\frac{3\pi}{4}}\; I_{2}}\end{pmatrix},} \right\}.}} & (67)\end{matrix}$

In this case, the W₂ codebook size is 4-bits.

Generalization

The precoders proposed above as alternative design 1 may be combinedwith the precoders in the previous sections to construct the finalrank-2 codebook. For example, assuming over-sampling ratio N (e.g.N=16), N_(b)=4, the rank-2 codebook may be expressed as

$\begin{matrix}{{{W_{1} \in C_{1}} = \left\{ {\begin{bmatrix}X^{(k)} & 0 \\0 & X^{(k)}\end{bmatrix} \cdot} \right\}},{i_{1} = {k = 0}},\ldots \;,{{2N\text{/}N_{b}} - 1}} & (68)\end{matrix}$

→size N/2

When i₂=0, . . . , 7;

$\begin{matrix}{X^{(k)} = \left\lfloor {b_{{({N_{b}{k/2}})}{mod}\mspace{11mu} N}\mspace{14mu} b_{{({{N_{b}{k/2}} + 1})}{mod}\; N}\mspace{14mu} \ldots \mspace{14mu} b_{{({{N_{b}{k/2}} + N_{b} - 1})}{mod}\mspace{11mu} N}} \right\rfloor} & (69) \\{{{W_{2} \in {CB}_{2}} = \left\{ {{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\Y_{1} & {- Y_{2}}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\{j\; Y_{1}} & {{- j}\; Y_{2}}\end{bmatrix}}} \right\}},} & (70) \\{\left( {Y_{1},Y_{2}} \right) \in \left\{ {\left( {e_{1}^{4},e_{1}^{4}} \right),\left( {e_{2}^{4},e_{2}^{4}} \right),\left( {e_{3}^{4},e_{3}^{4}} \right),\left( {e_{4}^{4},e_{4}^{4}} \right)} \right\}} & (71)\end{matrix}$

When i₂≦8, . . . , 15:

$\begin{matrix}{\mspace{79mu} {X^{(k)} = \left\lfloor {b_{{({k = {N/2}})}{mod}\mspace{11mu} N}\;,b_{k\; {mod}\; N}}\; \right\rfloor}} & (72) \\{{{W_{2} \in {CB}_{2}} = {\frac{1}{\sqrt{2}} \times \begin{Bmatrix}{\begin{bmatrix}Y_{1} & Y_{2} \\Y_{1} & Y_{2}\end{bmatrix},\begin{bmatrix}Y_{1} & Y_{2} \\Y_{1} & {- Y_{2}}\end{bmatrix},\begin{bmatrix}Y_{1} & Y_{2} \\{- Y_{1}} & Y_{2}\end{bmatrix},\begin{bmatrix}Y_{1} & Y_{2} \\{- Y_{1}} & {- Y_{2}}\end{bmatrix},} \\{\begin{bmatrix}Y_{1} & Y_{2} \\{j\; Y_{1}} & {j\; Y_{2}}\end{bmatrix},\begin{bmatrix}Y_{1} & Y_{2} \\{j\; Y_{1}} & {{- j}\; Y_{2}}\end{bmatrix},\begin{bmatrix}Y_{1} & Y_{2} \\{{- j}\; Y_{1}} & {j\; Y_{2}}\end{bmatrix},\begin{bmatrix}Y_{1} & Y_{2} \\{{- j}\; Y_{1}} & {{- j}\; Y_{2}}\end{bmatrix},}\end{Bmatrix}}},} & (73) \\{\mspace{79mu} {\left( {Y_{1},Y_{2}} \right) \in \left\{ \left( {e_{1}^{2},e_{2}^{2}} \right) \right\}}} & (74)\end{matrix}$

The W₁ overhead is log 2(N/2)=3-bits, and the W₂ overhead is 4-bits.

Note that the combined codebook can be re-expressed below.

$\begin{matrix}{{W_{l} \in C_{l}} = \left\{ {\begin{bmatrix}X^{(0)} & 0 \\0 & X^{(0)}\end{bmatrix},{\ldots \mspace{11mu}\begin{bmatrix}X^{({N - 1}} & 0 \\0 & X^{({N - 1})}\end{bmatrix}}} \right\}} & (75)\end{matrix}$

→size N

When

$\begin{matrix}{W_{l} = {\begin{bmatrix}X^{(k)} & 0 \\0 & X^{(k)}\end{bmatrix}\left( {{k = {_{1} = 0}},{{\ldots \mspace{14mu} N\text{/}2} - 1}} \right)\text{:}}} & (76) \\{X^{(k)} = \left\lfloor \begin{matrix}b_{{({N_{b}k\text{/}2})}{mod}\mspace{11mu} N} & b_{{({{N_{b}k\text{/}2} + 1})}{mod}\mspace{11mu} N} & \ldots & \left. b_{{({{N_{b}k\text{/}2} + N_{b} - 1})}{mod}\mspace{11mu} N} \right\rfloor\end{matrix} \right.} & (77) \\{{W_{2} \in {CB}_{2}} = \left\{ {{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\Y_{1} & {- Y_{2}}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\{j\; Y_{1}} & {{- j}\; Y_{2}}\end{bmatrix}}} \right\}} & (78) \\{\left( {Y_{1},Y_{2}} \right) \in \left\{ {\left( {e_{1}^{4},e_{1}^{4}} \right),\left( {e_{2}^{4},e_{2}^{4}} \right),\left( {e_{3}^{4},e_{3}^{4}} \right),\left( {e_{4}^{4},e_{4}^{4}} \right)} \right\}} & (79)\end{matrix}$

When

$\begin{matrix}{\mspace{79mu} {W_{1} = {\begin{bmatrix}X^{(k)} & 0 \\0 & X^{(k)}\end{bmatrix}\left( {{k = {_{1} = \frac{N}{2}}},\ldots \mspace{14mu},{N - 1}} \right)\text{:}}}} & (80) \\{\mspace{79mu} {X^{(k)} = \left\lfloor {b_{{({k - {N\text{/}2}})}{mod}\mspace{11mu} N},b_{k\mspace{11mu} {mod}\mspace{11mu} N}} \right\rfloor}} & (81) \\{{{W_{2} \in {CB}_{2}} = {\frac{1}{\sqrt{2}} \times \begin{Bmatrix}{\begin{bmatrix}Y_{1} & Y_{2} \\Y_{1} & Y_{2}\end{bmatrix},\begin{bmatrix}Y_{1} & Y_{2} \\Y_{1} & {- Y_{2}}\end{bmatrix},\begin{bmatrix}Y_{1} & Y_{2} \\{- Y_{1}} & Y_{2}\end{bmatrix},\begin{bmatrix}Y_{1} & Y_{2} \\{- Y_{1}} & {- Y_{2}}\end{bmatrix},} \\{\begin{bmatrix}Y_{1} & Y_{2} \\{j\; Y_{1}} & {j\; Y_{2}}\end{bmatrix},\begin{bmatrix}Y_{1} & Y_{2} \\{j\; Y_{1}} & {{- j}\; Y_{2}}\end{bmatrix},\begin{bmatrix}Y_{1} & Y_{2} \\{{- j}\; Y_{1}} & {j\; Y_{2}}\end{bmatrix},\begin{bmatrix}Y_{1} & Y_{2} \\{{- j}\; Y_{1}} & {{- j}\; Y_{2}}\end{bmatrix},}\end{Bmatrix}}},} & (82) \\{\mspace{79mu} {\left( {Y_{1},Y_{2}} \right) \in \left\{ \left( {e_{1}^{2},e_{2}^{2}} \right) \right\}}} & (83)\end{matrix}$

In this case, the W₁ overhead is log 2(N)=4-bits, and the W₂ overhead is3-bits.

Another possible combinatorial design is

$\begin{matrix}{{W_{1} \in C_{1}} = \left\{ {\begin{bmatrix}X^{(0)} & 0 \\0 & X^{(0)}\end{bmatrix},{\ldots \mspace{14mu}\begin{bmatrix}X^{({N - 1}} & 0 \\0 & X^{({N - 1})}\end{bmatrix}}} \right\}} & (84)\end{matrix}$

→size N

When

$\begin{matrix}{\mspace{79mu} {W_{1} = {\begin{bmatrix}X^{(k)} & 0 \\0 & X^{(k)}\end{bmatrix}\left( {{k = {_{1} = 0}},{{\ldots \mspace{14mu} N\text{/}2} - 1}} \right)\text{:}}}} & (85) \\{\mspace{79mu} {X^{(k)} = \left\lfloor \begin{matrix}b_{{({N_{b}k\text{/}2})}{mod}\mspace{11mu} N} & b_{{({{N_{b}k\text{/}2} + 1})}{mod}\mspace{11mu} N} & \ldots & \left. b_{{({{N_{b}k\text{/}2} + N_{b} - 1})}{mod}\mspace{11mu} N} \right\rfloor\end{matrix} \right.}} & (86) \\{\mspace{79mu} {{{W_{2} \in {CB}_{2}} = \left\{ {{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\Y_{1} & {- Y_{2}}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\{j\; Y_{1}} & {- {jY}_{2}}\end{bmatrix}}} \right\}},}} & (87) \\{\left( {Y_{1},Y_{2}} \right) \in \left\{ {\left( {e_{1}^{4},e_{1}^{4}} \right), \left( {e_{2}^{4}, e_{2}^{4}} \right), \left( {e_{3}^{4}, e_{3}^{4}} \right), \left( {e_{4}^{4}, e_{4}^{4}} \right), \left( {e_{1}^{4},e_{2}^{4}} \right), \left( {e_{2}^{4},e_{3}^{4}} \right), \left( {e_{1}^{4},e_{4}^{4}} \right),\left( {e_{2}^{4},e_{4}^{4}} \right)} \right\}} & (88)\end{matrix}$

When

$\begin{matrix}{\mspace{79mu} {W_{1} = {\begin{bmatrix}X^{(k)} & 0 \\0 & X^{(k)}\end{bmatrix}\left( {{k = {_{1} = \frac{N}{2}}},\ldots \mspace{14mu},{N - 1}} \right)\text{:}}}} & (89) \\{\mspace{79mu} {X^{(k)} = \left\lfloor {b_{{({k = {N\text{/}2}})}{mod}\mspace{11mu} N},b_{k\mspace{11mu} {mod}\mspace{11mu} N}} \right\rfloor}} & (90) \\{{{W_{2} \in {CB}_{2}} = {\frac{1}{\sqrt{2}} \times \begin{Bmatrix}\begin{matrix}\begin{matrix}{\begin{bmatrix}Y_{1} & Y_{2} \\Y_{1} & Y_{2}\end{bmatrix},\begin{bmatrix}Y_{1} & Y_{2} \\Y_{1} & {- Y_{2}}\end{bmatrix},\begin{bmatrix}Y_{1} & Y_{2} \\{- Y_{1}} & Y_{2}\end{bmatrix},\begin{bmatrix}Y_{1} & Y_{2} \\{- Y_{1}} & {- Y_{2}}\end{bmatrix},} \\{\begin{bmatrix}Y_{1} & Y_{2} \\{j\; Y_{1}} & {j\; Y_{2}}\end{bmatrix},\begin{bmatrix}Y_{1} & Y_{2} \\{j\; Y_{1}} & {{- j}\; Y_{2}}\end{bmatrix},\begin{bmatrix}Y_{1} & Y_{2} \\{{- j}\; Y_{2}} & {j\; Y_{2}}\end{bmatrix},\begin{bmatrix}Y_{1} & Y_{2} \\{{- j}\; Y_{1}} & {{- j}\; Y_{2}}\end{bmatrix},}\end{matrix} \\{\begin{bmatrix}Y_{1} & Y_{2} \\{^{j\; \frac{\pi}{4}}Y_{1}} & {^{j\; \frac{\pi}{4}}Y_{2}}\end{bmatrix},\begin{bmatrix}Y_{1} & Y_{2} \\{^{j\; \frac{\pi}{4}}Y_{1}} & {{- ^{j\; \frac{\pi}{4}}}Y_{2}}\end{bmatrix},\begin{bmatrix}Y_{1} & Y_{2} \\{{- ^{j\; \frac{\pi}{4}}}Y_{1}} & {^{j\; \frac{\pi}{4}}Y_{2}}\end{bmatrix},\begin{bmatrix}Y_{1} & Y_{2} \\{{- ^{j\; \frac{\pi}{4}}}Y_{1}} & {{- ^{j\; \frac{\pi}{4}}}Y_{2}}\end{bmatrix},}\end{matrix} \\{\begin{bmatrix}Y_{1} & Y_{2} \\{^{j\frac{3\; \pi}{4}}Y_{1}} & {^{j\; \frac{3\; \pi}{4}}Y_{2}}\end{bmatrix},\begin{bmatrix}Y_{1} & Y_{2} \\{^{j\frac{3\; \pi}{4}}Y_{1}} & {{- ^{j\frac{3\; \pi}{4}}}Y_{2}}\end{bmatrix},\begin{bmatrix}Y_{1} & Y_{2} \\{{- ^{j\frac{3\; \pi}{4}}}Y_{1}} & {^{j\frac{3\; \pi}{4}}Y_{2}}\end{bmatrix},\begin{bmatrix}Y_{1} & Y_{2} \\{{- ^{j\frac{3\; \pi}{4}}}Y_{1}} & {{- ^{j\frac{3\; \pi}{4}}}Y_{2}}\end{bmatrix},}\end{Bmatrix}}},} & (91) \\{\mspace{79mu} {\left( {Y_{1},Y_{2}} \right) \in \left\{ \left( {e_{1}^{2},e_{2}^{2}} \right) \right\}}} & (92)\end{matrix}$

The W₁ overhead is log 2(N)=4-bits, and the W₂ overhead is 4-bits.

Inheritance of the Release 8 Codebook in Release 12

Using Release 8 Precoders for W₂

As proposed in the previous sections, it is possible for the Release 12codebook to inherit the Release 8 codebook as a subset. This can beachieved by using the Release 8 4Tx precoder as the codebook for W₂which is associated with W₁=4×4 identity matrix I₄. This is applicablefor rank-1 to rank-4, where the subband W₂ overhead is 4-bits persubband.

It is also possible to divide the Release 8 precoders into N groups,where each group has 16/N Release 8 precoders that form a different W₂codebook. For each W₂ codebook, the W₁ matrix is equivalent to a 4×4identity matrix. The subband W₂ overhead will therefore be reduced tolog 2(16/N) bits.

For example:

-   -   The Release 8 codebook is inherited in Release 12 codebook,        corresponding to two W₁ matrices, e.g., W₁ ^(k)=I₄, k=0, 1, For        W₁ ⁰, the W₂ codebook comprises the first eight Release 8        precoders. For W₁ ¹, the W₂ codebook comprises the last eight        Release 8 precoders.    -   Other values of N are possible to adapt the subband PMI        bit-width. For example, N=2 corresponds to a subband size of log        2(8)=3-bits, N=4 corresponds to a subband size of log 2(4)=2        bits.    -   The above designs can be applied for rank-1, rank-2, rank-3, and        rank-4.

Constructing W₁ with Release 8 Precoders and W₂ with Column Selection

It is further possible to inherit the Release 8 codebook in Release 12by constructing the W₁ matrix with Release 8 precoders, and constructingthe W₂ codebook with column selection matrices.

For example:

-   -   For rank-1, W₁ includes all or a subset of Release 8 rank-1        vectors. The size of W₁ is given by 4×L, where 1<=L<=16 is the        number of rank-1 Release 8 codebook vectors that are included in        W₁. The W₂ codebook comprises L column-selection vectors [e₁, e₂        . . . e_(L)], where e_(i) is an L×1 vector of all zero entries,        except for the i-th entry, which is equivalent to 1.    -   For rank-r (r=2, 3, 4), the columns of W₁ comprise all or a        subset of Release 8 rank-r vectors, e.g. this may be denoted as

W ₁ =└W _(s(0)) ^(Rel.8-rank-r) W _(s(1)) ^(Rel.8-rank-r) . . . W_(s(L-1)) ^(Rel.8-rank-r)┘  (93)

where W_(s(1)) ^(Rel.8-rank-r) is the s(1)-th Release 8 precoder. Thesize of W₁ is 4×rL, where 1<=L<=16 is the number of rank-r Release 8codebook matrices in W₁. The W₂ codebook comprises L column-selectionmatrices, where the I-th W₂ matrix (1<=1<=L) is

$\begin{matrix}{{W\; 2} = \begin{bmatrix}0_{{r{({t - 1})}} \times r} \\I_{r \times r} \\0_{{r{({L - 1})}} \times r}\end{bmatrix}} & (94)\end{matrix}$

Alternatively, the W₁ matrix can be constructed in a block diagonalmanner.

For example, for rank-1:

$\begin{matrix}{{W_{1} = \begin{bmatrix}A & 0 \\0 & B\end{bmatrix}},} & (95)\end{matrix}$

where A and B are of sizes 2×16, where the l-th column of A is the firsttwo rows of the l-th Release 8 precoder, and the l-th column of B is thelast two rows of the l-th Release 8 precoder, l=1, . . . , 16.

The W2 matrix can be written in the form of

$\begin{matrix}{{W_{2}^{k} = \begin{bmatrix}e_{k} & 0 \\0 & e_{k}\end{bmatrix}},{k = 1},\ldots \mspace{14mu},16,} & (96)\end{matrix}$

where e_(k) is the k-th column of a 16×16 identity matrix.

For rank-r, r=2, . . . , 4:

$\begin{matrix}{{W_{1} = \begin{bmatrix}A & 0 \\0 & B\end{bmatrix}},} & (97)\end{matrix}$

where A and B are of sizes 2×16r, and where

A=└W₁ ^(Rel.8-rank-r) W₂ ^(Rel.8-rank-r) . . . W₁₆^(Rel.8-rank-r)┘_((1:2)), P_((1:2)) is the first and second row ofmatrix P, and

B=└W₁ ^(Rel.8-rank-r) W₂ ^(Rel.8-rank-r) . . . W₁₆^(Rel.8-rank-r)┘_((3:4)), P_((3:4)) is the third and fourth row ofmatrix P.

For the W₂ codebook,

$\begin{matrix}{{W_{2}^{k} = \begin{bmatrix}e_{k} & 0 \\0 & e_{k}\end{bmatrix}},{k = 1},\ldots \mspace{14mu},16,{{{where}\mspace{14mu} e_{k}} = {\begin{bmatrix}0_{{r{({k - 1})}} \times r} \\I_{r \times r} \\0_{{r{({L - k})}} \times r}\end{bmatrix}.}}} & (98)\end{matrix}$

4Tx Codebook Enhancements for LTE

In the following section, possible 4Tx codebook enhancement alternativesfor LTE Release 12 are disclosed. In these examples, {tilde over(e)}_(i) is a 4×1 vector with all zero entries except for the i-thelement which has value 1.

Rank-1/2

For a rank-1/2 codebook, the following two alternatives are possible.

Alt-1:

Reusing the 8Tx GoB design with N=32 over-sampled beams and 4 adjacentbeams per grid, the following 4Tx codebook can be considered.

$\begin{matrix}{{B = \begin{bmatrix}b_{0} & b_{1} & \ldots & b_{31}\end{bmatrix}}, {\lbrack B\rbrack_{{1 + m},{1 + n}} = ^{j\frac{2\; {mn}}{32}}}, {m = 0}, {{\ldots \mspace{14mu} \frac{N_{1}}{2}} - 1}, {n = 0},1,\ldots \mspace{14mu},31} & (99) \\{X^{(k)} \in \left\{ {{{\left\lfloor \begin{matrix}b_{2\; k\mspace{11mu} {mod}\mspace{11mu} 32} & b_{{({{2k} + 1})}\mspace{11mu} {mod}\mspace{11mu} 32} & b_{{({{2k} + 2})}\mspace{11mu} {mod}\mspace{11mu} 32} & b_{{({{2k} + 3})}\mspace{11mu} {mod}\mspace{11mu} 32}\end{matrix} \right\rfloor \text{:}\mspace{14mu} k} = 0}, 1, \ldots \mspace{14mu},15} \right\}} & (100) \\{\mspace{79mu} {W_{l}^{(k)} = \begin{bmatrix}X^{(k)} & 0 \\0 & X^{(k)}\end{bmatrix}}} & (101)\end{matrix}$Codebook1:C ₁ ={W ₁ ⁽⁰⁾ ,W ₁ ⁽¹⁾ ,W ₁ ⁽²⁾ , . . . ,W ₁ ⁽¹⁵⁾}  (102)

Rank-1: (4-bit)

$\begin{matrix}{{W_{2} \in C_{2}} = \left\{ {{\frac{1}{\sqrt{2}}\begin{bmatrix}Y \\Y\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y \\{j\; Y}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y \\{- Y}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y \\{{- j}\; Y}\end{bmatrix}}} \right\}} & (103) \\{Y \in \left\{ {{\overset{\sim}{e}}_{1},{\overset{\sim}{e}}_{2},{\overset{\sim}{e}}_{3},{\overset{\sim}{e}}_{4}} \right\}} & (104)\end{matrix}$

Rank-2: (4-bit)

$\begin{matrix}{\mspace{79mu} {{W_{2} \in C_{2}} = \left\{ {{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\Y_{1} & {- Y_{2}}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\{j\; Y_{1}} & {{- j}\; Y_{2}}\end{bmatrix}}} \right\}}} & (105) \\{\left( {Y_{1},Y_{2}} \right) \in \left\{ {\left( {{\overset{\sim}{e}}_{1},{\overset{\sim}{e}}_{1}} \right), \left( {{\overset{\sim}{e}}_{2}, {\overset{\sim}{e}}_{2}} \right), \left( {{\overset{\sim}{e}}_{3}, {\overset{\sim}{e}}_{3}} \right), \left( {{\overset{\sim}{e}}_{4}, {\overset{\sim}{e}}_{4}} \right), \left( {{\overset{\sim}{e}}_{1},{\overset{\sim}{e}}_{2}} \right), \left( {{\overset{\sim}{e}}_{2},{\overset{\sim}{e}}_{3}} \right), \left( {{\overset{\sim}{e}}_{1},{\overset{\sim}{e}}_{4}} \right),\left( {{\overset{\sim}{e}}_{2},{\overset{\sim}{e}}_{4}} \right)} \right\}} & (106)\end{matrix}$

Rank-2: (3-bit)

For rank-2, if a (3-bit) W₂ is preferred, (Y1, Y2) can be changed to:

(Y ₁ ,Y ₂)ε{({tilde over (e)} ₁ ,{tilde over (e)} ₁),({tilde over (e)} ₂,{tilde over (e)} ₂),({tilde over (e)} ₃ ,{tilde over (e)} ₃),({tildeover (e)} ₄ ,{tilde over (e)} ₄)}.  (107)

Each W1 matrix is constructed by a set of adjacent DFT beams and coversa narrow range of angle-of-departures/arrivals. The fundamentalconsideration is based the feedback of wideband/long-term channel basedon W₁ matrix, and the fed back W1 matrix in a properly designed C1codebook should be able to reflect the wideband AoA/AoDs with sufficientaccuracy. For instance, most macro base stations in cellularcommunication systems are elevated on a cell tower and likely have adirect line-of-sign to the UE, where the angle of incoming radio signalsto a UE is within in a small range. Hence, the wideband W1 comprising aset of adjacent beams can be used to cover the range of incoming radiosignals on the wideband, whereas the narrowband W2 codebook can be usedto select a specific beam on each subband. This W₁ design is particularsuitable to macro base station with narrowly spaced antennas,propagation channel with sufficient line-of-sights, and perfectlycalibrated base station antennas.

Alt-2:

A rank-1/2 codebook comprises two components, where W₁ structures aredifferent in each component. For the first 8 W₁ matrices, Xn comprisefour adjacent DFT beams with over-sampling rate of N=16. For the last 8W₁ matrices, Xn comprise four distributed non-adjacent DFT beamsuniformly sampling the [0, 360] angle of arrival sub-space. Thisprovides wider angular spread coverage and may be beneficial to largetiming misalignment error.

The W₁ codebook therefore can be given by:

$\begin{matrix}{{_{1} = 0},1,\ldots \mspace{14mu}, {{7\text{:}\mspace{14mu} X^{(i_{1})}} \in \left\{ \left\lfloor \begin{matrix}b_{2\; i_{1}{mod}\mspace{11mu} 16} & b_{{({{2\; i_{1}} + 1})}{mod}\mspace{11mu} 16} & b_{{({{2\; i_{1}} + 2})}{mod}\mspace{11mu} 16} & b_{{({{2\; i_{1}} + 3})}{mod}\mspace{11mu} 16}\end{matrix} \right\rfloor \right\}},} & (108) \\{\mspace{79mu} {{{b_{n}\left( {m + 1} \right)} = ^{j\; \frac{2{nmn}}{16}}},{n = 0},1,\ldots \mspace{14mu},15,{m = 0},{.1}}} & (109) \\{{_{1} = 8},9,\ldots \mspace{14mu}, {{15\text{:}\mspace{14mu} X^{(i_{2})}} \in \left\{ \left\lfloor \begin{matrix}b_{{({i_{1} - 8})}{mod}\mspace{11mu} 32} & b_{{({i_{1} - 8})} + {8\; {mod}\mspace{11mu} 32}} & b_{{(\; {i_{1} - 8})} + {16\; {mod}\mspace{11mu} 32}} & b_{{(\; {i_{1} - 8})} + 2 + {{mod}\mspace{11mu} 32}}\end{matrix} \right\rfloor \right\}},} & (110) \\{\mspace{79mu} {{{b_{n}\left( {m + 1} \right)} = ^{j\; \frac{2{nmn}}{32}}},{n = 0},1,\ldots \mspace{14mu},31,{m = 0},{.1}}} & (111)\end{matrix}$

As can be seen, for the last eight W matrices, each W₁ matrix incodebook C₁ is comprised of four non-adjacent DFT beams. The four DFTbeams in each W₁ matrix are widely spaced and uniformly distributed inthe [0,360] DFT subspace to cover a wide range ofangle-of-arrivals/departures. The DFT beams in a first W1 matrix arerotated by a small angle against four DFT beams in a second W₁ matrix.More specifically, the W1 matrices comprising widely spaced DFT beamsare summarized as

$\begin{matrix}{\begin{matrix}{X_{1}^{(8)} \in \left\{ \left\lfloor \begin{matrix}b_{0} & b_{8} & b_{16} & b_{24}\end{matrix} \right\rfloor \right\}} \\{X_{1}^{(9)} \in \left\{ \left\lfloor \begin{matrix}b_{1} & b_{9} & b_{17} & b_{25}\end{matrix} \right\rfloor \right\}} \\\ldots \\{X_{1}^{(15)} \in \left\{ \left\lfloor \begin{matrix}b_{7} & b_{15} & b_{23} & b_{31}\end{matrix} \right\rfloor \right\}}\end{matrix}\quad} & (112)\end{matrix}$

Such a design framework is particularly beneficial in uses cases (e.g.,widely spaced antenna components, un-calibrated antenna array, richmultipath scattering environment) where the angle-of-arrival/departuresof the incoming wireless signals are distributed in a wide range. Forinstance, in a heterogeneous deployment scenario where dense small cellsare overlaid on top of macro base station on the same frequency, themultipath radio signals received by a UE are reflected by numerousscattering objects surrounding the UE (e.g., building, cars). Havewidely spaced and non-adjacent DFT beams in W1 ensures that all incomingsignals from different angles can be properly captured. As another usecase, it is noted that the transmitter timing at different antennas on abase station shall be synchronized by careful timing alignmentcalibration. In practice, perfect timing alignment cannot always beguaranteed at a base station, especially for low-cost low-power smallbase stations (e.g. pico-cells, femto-cells) with cheaper RF components.In 3GPP LTE, a maximum 65 nano-second downlink transmission timingmisalignment requirement is set forth for all base stations. Aconsequence of misaligned antenna timing in the time domain is theincreased channel variation on different OFDM subcarriers in thefrequency domain, and the main DFT beam angle on one frequency subbandcan be significantly different than the main DFT beam angle on anothersubband. In this case, having non-adjacent widely spaced DFT beams inthe W₁ matrix ensures that the wideband angle-of-arrival/departures canbe more reliably covered, resulting in higher feedback accuracy.

The W₂ codebook for rank-1 (4-bit) may be given by:

$\begin{matrix}{{{W_{2} \in C_{2}} = \left\{ {{\frac{1}{\sqrt{2}}\begin{bmatrix}Y \\Y\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y \\{j\; Y}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y \\{- Y}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y \\{{- j}\; Y}\end{bmatrix}}} \right\}},} & (113) \\{Y \in \left\{ {{\overset{\sim}{e}}_{1},{\overset{\sim}{e}}_{2},{\overset{\sim}{e}}_{3},{\overset{\sim}{e}}_{4}} \right\}} & (114)\end{matrix}$

The W₂ codebook for rank-2 (4-bit) may be given by:

For W₂ corresponding to i₁=0, 1, . . . , 7

$\begin{matrix}{\mspace{79mu} {{{W_{2} \in C_{2}} = \left\{ {{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\Y_{1} & {- Y_{2}}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\{j\; Y_{1}} & {{- j}\; Y_{2}}\end{bmatrix}}} \right\}},}} & (115) \\{\left( {Y_{1},Y_{2}} \right) \in \left\{ {\left( {{\overset{\sim}{e}}_{1},{\overset{\sim}{e}}_{1}} \right),\left( {{\overset{\sim}{e}}_{2},{\overset{\sim}{e}}_{2}} \right),\left( {{\overset{\sim}{e}}_{3},{\overset{\sim}{e}}_{3}} \right),\left( {{\overset{\sim}{e}}_{4},{\overset{\sim}{e}}_{4}} \right),\left( {{\overset{\sim}{e}}_{1},{\overset{\sim}{e}}_{2}} \right),\left( {{\overset{\sim}{e}}_{2},{\overset{\sim}{e}}_{3}} \right),\left( {{\overset{\sim}{e}}_{1},{\overset{\sim}{e}}_{4}} \right),\left( {{\overset{\sim}{e}}_{2},{\overset{\sim}{e}}_{4}} \right)} \right\}} & (116)\end{matrix}$

If a (3-bit) W₂ is preferred, (Y1, Y2) may be changed to:

(Y ₁ ,Y ₂)ε{({tilde over (e)} ₁ ,{tilde over (e)} ₁),({tilde over (e)} ₂,{tilde over (e)} ₂),({tilde over (e)} ₃ ,{tilde over (e)} ₃),({tildeover (e)} ₄ ,{tilde over (e)} ₄)}.  (117)

For W₂ corresponding to i₁=8, 9, . . . , 15

$\begin{matrix}{{W_{2} \in C_{2}} = \begin{Bmatrix}{{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\Y_{1} & Y_{2}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\Y_{1} & {- Y_{2}}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\{- Y_{1}} & Y_{2}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\{- Y_{1}} & {- Y_{2}}\end{bmatrix}},} \\{{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\{j\; Y_{1}} & {j\; Y_{2}}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\{j\; Y_{1}} & {{- j}\; Y_{2}}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\{{- j}\; Y_{1}} & {j\; Y_{2}}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\{{- j}\; Y_{1}} & {{- j}\; Y_{2}}\end{bmatrix}},}\end{Bmatrix}} & (118) \\{\left( {Y_{1},Y_{2}} \right) \in \left\{ {\left( {{\overset{\sim}{e}}_{1},{\overset{\sim}{e}}_{3}} \right),\left( {{\overset{\sim}{e}}_{2},{\overset{\sim}{e}}_{4}} \right)} \right\}} & (119)\end{matrix}$

If a (3-bit) W₂ is preferred, the W₂ codebook may be changed to

$\begin{matrix}{{W_{2} \in C_{2}} = \left\{ {{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\Y_{1} & Y_{2}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\Y_{1} & {- Y_{2}}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\{- Y_{1}} & Y_{2}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\{- Y_{1}} & {- Y_{2}}\end{bmatrix}}} \right\}} & (120) \\{{\left( {Y_{1},Y_{2}} \right) \in \left\{ {\left( {{\overset{\sim}{e}}_{1},{\overset{\sim}{e}}_{3}} \right),\left( {{\overset{\sim}{e}}_{2},{\overset{\sim}{e}}_{4}} \right)} \right\}},} & (121) \\{or} & \; \\{{W_{2} \in C_{2}} = \begin{Bmatrix}{{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\Y_{1} & Y_{2}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\Y_{1} & {- Y_{2}}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\{- Y_{1}} & Y_{2}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\{- Y_{1}} & {- Y_{2}}\end{bmatrix}},} \\{{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\{j\; Y_{1}} & {j\; Y_{2}}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\{j\; Y_{1}} & {{- j}\; Y_{2}}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\{{- j}\; Y_{1}} & {j\; Y_{2}}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\{{- j}\; Y_{1}} & {{- j}\; Y_{2}}\end{bmatrix}},}\end{Bmatrix}} & (122) \\{\left( {Y_{1},Y_{2}} \right) \in {\left\{ \left( {{\overset{\sim}{e}}_{1},{\overset{\sim}{e}}_{3}} \right) \right\} \mspace{14mu} {or}\mspace{14mu} \left\{ \left( {{\overset{\sim}{e}}_{2},{\overset{\sim}{e}}_{4}} \right) \right\}}} & (123)\end{matrix}$

In one embodiment, the new rank-1/2 codebook designs set forth above maybe incorporated into an LTE system complying with LTE Release 12. Asubset of precoders in the proposed codebooks above may be used toconstruct a new Release 12 4Tx codebook, for either rank-1 or rank-2.For example, the W1 codebook may be constructed by W₁ matrices onlycomprising widely spaced DFT beams, e.g., i₂=8, 9, . . . 15 in Equation(110)-(111).

FIG. 6 illustrates the precoding matrix/vector selection processaccording to one embodiment. The final precoding matrix/vector is afunction of two PMIs:

W=ƒ(PMI ₁ ,PMI ₂)  (124)

where PMI₁ is updated at a significantly less frequent rate than PMI₂.PMI₁ is intended for the entire system bandwidth while PMI₂ can befrequency-selective.

FIG. 6 illustrates the technique used in downlink LTE-Advanced (LTE-A).The UE selects PMI₁ and PMI₂ and hence W₁ and W₂ in a manner similar tothe LTE feedback paradigm.

The UE first selects the first precoder codebook W₁ in block 601 basedon the long-term channel properties such as spatial covariance matrixsuch as in a spatial correlation domain from an input of PMI₁. This isdone in a long-term basis consistent with the fact that spatialcovariance matrix needs to be estimated over a long period of time andin a wideband manner.

Conditioned upon W₁, the UE selects W₂ based on the short-term(instantaneous) channel. This is a two stage process. In block 602, aset of codebooks CB₂ ⁽⁰⁾ to CB₂ ^((N-1)) is selected based upon the PMI₁input. Block 603 selects one precoder corresponding to the selectedcodebook CB₂ ^((PMI) ¹ ⁾ and PMI₂. This selection may be conditionedupon the selected rank indicator (RI). Alternatively, RI can be selectedjointly with W₂. Block 604 takes the selected W₁ and W₂ forms thefunction ƒ(W₁, W₂).

PMI₁ and PMI₂ are reported to the base station (eNodeB) at differentrates and/or different frequency resolutions.

Based on this design framework, several types of codebook design aredescribed herein. While each type can stand alone, it is also possibleto use different types in a single codebook design especially if thedesign is intended for different scenarios. A simple yet versatiledesign can be devised as follows:

PMI₁ selects one of the N codebooks W₁ as indicated above.

PMI₂ selects at least one of the column vectors of W, wherein the numberof selected column vectors is essentially the recommended transmissionrank (RI).

This design allows construction of N different scenarios where thecodebook W₁ for each scenario is chosen to contain a set of basisvectors for a particular spatial channel characteristic W₂. While anytwo-dimensional function can be used in equation (124), this disclosureassumes a product (matrix multiplication) function f(x,y)=xy. Thus thefinal short-term precoding matrix/vector is computed as a matrix productof W₁ and W₂: W=W₁W₂.

FIG. 7 is a block diagram illustrating internal details of a mobile UE701 and an eNodeB 702 in the network system of FIG. 1. Mobile UE 701 mayrepresent any of a variety of devices such as a server, a desktopcomputer, a laptop computer, a cellular phone, a Personal DigitalAssistant (PDA), a smart phone or other electronic devices. In someembodiments, the electronic mobile UE 701 communicates with eNodeB 702based on a LTE or Evolved Universal Terrestrial Radio Access Network(E-UTRAN) protocol. Alternatively, another communication protocol nowknown or later developed can be used.

Mobile UE 701 comprises a processor 703 coupled to a memory 704 and atransceiver 705. The memory 704 stores (software) applications 706 forexecution by the processor 703. The applications could comprise anyknown or future application useful for individuals or organizations.These applications could be categorized as operating systems (OS),device drivers, databases, multimedia tools, presentation tools,Internet browsers, emailers, Voice-Over-Internet Protocol (VOIP) tools,file browsers, firewalls, instant messaging finance tools, games, wordprocessors or other categories. Regardless of the exact nature of theapplications, at least some of the applications may direct the mobile UE701 to transmit UL signals to eNodeB (base-station) 702 periodically orcontinuously via the transceiver 705. In at least some embodiments, themobile UE 701 identifies a Quality of Service (QoS) requirement whenrequesting an uplink resource from eNodeB 702. In some cases, the QoSrequirement may be implicitly derived by eNodeB 702 from the type oftraffic supported by the mobile UE 701.

As an example, VOIP and gaming applications often involve low-latencyuplink (UL) transmissions while High Throughput (HTP)/HypertextTransmission Protocol (HTTP) traffic can involve high-latency uplinktransmissions.

Transceiver 705 includes uplink logic which may be implemented byexecution of instructions that control the operation of the transceiver.Some of these instructions may be stored in memory 704 and executed whenneeded by processor 703. As would be understood by one of skill in theart, the components of the uplink logic may involve the physical (PHY)layer and/or the Media Access Control (MAC) layer of the transceiver705. Transceiver 705 includes one or more receivers 707 and one or moretransmitters 708.

Processor 703 may send or receive data to various input/output devices709. A subscriber identity module (SIM) card stores and retrievesinformation used for making calls via the cellular system. A Bluetoothbaseband unit may be provided for wireless connection to a microphoneand headset for sending and receiving voice data. Processor 703 may sendinformation to a display unit for interaction with a user of mobile UE701 during a call process. The display may also display picturesreceived from the network, from a local camera, or from other sourcessuch as a Universal Serial Bus (USB) connector. Processor 703 may alsosend a video stream to the display that is received from various sourcessuch as the cellular network via RF transceiver 705 or the camera.

During transmission and reception of voice data or other applicationdata, transmitter 707 may be or become non-synchronized with its servingeNodeB. In this case, it sends a random access signal. As part of thisprocedure, it determines a preferred size for the next datatransmission, referred to as a message, by using a power threshold valueprovided by the serving eNodeB, as described in more detail above. Inthis embodiment, the message preferred size determination is embodied byexecuting instructions stored in memory 704 by processor 703. In otherembodiments, the message size determination may be embodied by aseparate processor/memory unit, by a hardwired state machine, or byother types of control logic, for example.

eNodeB 702 comprises a processor 710 coupled to a memory 711, symbolprocessing circuitry 712, and a transceiver 713 via backplane bus 714.The memory stores applications 715 for execution by processor 710. Theapplications could comprise any known or future application useful formanaging wireless communications. At least some of the applications 715may direct eNodeB 702 to manage transmissions to or from mobile UE 701.

Transceiver 713 comprises an uplink Resource Manager, which enableseNodeB 702 to selectively allocate uplink Physical Uplink Shared CHannel(PUSCH) resources to mobile UE 701. As would be understood by one ofskill in the art, the components of the uplink resource manager mayinvolve the physical (PHY) layer and/or the Media Access Control (MAC)layer of the transceiver 713. Transceiver 713 includes at least onereceiver 715 for receiving transmissions from various UEs within rangeof eNodeB 702 and at least one transmitter 716 for transmitting data andcontrol information to the various UEs within range of eNodeB 702.

The uplink resource manager executes instructions that control theoperation of transceiver 713. Some of these instructions may be locatedin memory 711 and executed when needed on processor 710. The resourcemanager controls the transmission resources allocated to each UB 701served by eNodeB 702 and broadcasts control information via the PDCCH.

Symbol processing circuitry 712 performs demodulation using knowntechniques. Random access signals are demodulated in symbol processingcircuitry 712.

During transmission and reception of voice data or other applicationdata, receiver 715 may receive a random access signal from a UE 701. Therandom access signal is encoded to request a message size that ispreferred by UE 701. UE 701 determines the preferred message size byusing a message threshold provided by eNodeB 702. In this embodiment,the message threshold calculation is embodied by executing instructionsstored in memory 711 by processor 710. In other embodiments, thethreshold calculation may be embodied by a separate processor/memoryunit, by a hardwired state machine, or by other types of control logic,for example. Alternatively, in some networks the message threshold is afixed value that may be stored in memory 711, for example. In responseto receiving the message size request, eNodeB 702 schedules anappropriate set of resources and notifies UE 701 with a resource grant.

Many modifications and other embodiments of the invention(s) will cometo mind to one skilled in the art to which the invention(s) pertainhaving the benefit of the teachings presented in the foregoingdescriptions, and the associated drawings. Therefore, it is to beunderstood that the invention(s) are not to be limited to the specificembodiments disclosed. Although specific terms are employed herein, theyare used in a generic and descriptive sense only and not for purposes oflimitation.

1-48. (canceled)
 49. A method of channel state information (CSI)feedback in a wireless communication system, node comprising: receivingone or more precoding matrix indicator (PMI) signals from a remotetransceiver; and generating a precoding matrix W derived from a matrixmultiplication of two matrices W1 and W2, where said precoding matrix Wis applicable to precoding one or more layers of data streams, matrix W1is selected from a first codebook C₁ based on a first group of bits inthe PMI signals, and matrix W2 is selected from a second codebook C₂based on a second group of bits in the PMI signals, wherein the firstcodebook C₁ comprises the following W1 matrices constructed bynon-adjacent Discrete Fourier Transform (DFT) vectors:$B = \begin{bmatrix}b_{0} & b_{1} & \ldots & b_{31}\end{bmatrix}$ ${b_{k} = \begin{bmatrix}1 & ^{{j2\pi}\; {k/32}}\end{bmatrix}^{T}},{for}$ k = 0, 1, …  , 31${X^{(k)} = \begin{bmatrix}b_{k} & b_{{({k + 8})}{mod}\; 32} & b_{{({k + 16})}{mod}\; 32} & b_{{({k + 24})}{mod}\; 32}\end{bmatrix}},{for}$ k = 0, 1, …  , 15 $W_{1}^{(k)} = \begin{bmatrix}X^{(k)} & 0 \\0 & X^{(k)}\end{bmatrix}$ for k = 0, 1, …  , 15 C₁ = {W₁⁽⁰⁾, W₁⁽¹⁾, …  , W₁⁽¹⁵⁾}for k = 0, 1, …  ,
 15. 50. The method of claim 49, wherein the numberof layers of the data stream is one, and wherein the second codebook C₂comprises at least one matrix of the following form: $\begin{matrix}{{{W_{2} \in C_{2}} = \left\{ {{\frac{1}{2}\begin{bmatrix}Y \\Y\end{bmatrix}},{\frac{1}{2}\begin{bmatrix}Y \\{j\; Y}\end{bmatrix}},{\frac{1}{2}\begin{bmatrix}Y \\{- Y}\end{bmatrix}},{\frac{1}{2}\begin{bmatrix}Y \\{{- j}\; Y}\end{bmatrix}}} \right\}},{Y \in \left\{ {{\overset{\sim}{e}}_{1},{\overset{\sim}{e}}_{2},{\overset{\sim}{e}}_{3},{\overset{\sim}{e}}_{4}} \right\}}} & \;\end{matrix}$ where {tilde over (e)}₁ is a 4×1 vector with all zeroentries except for the i-th element which has value
 1. 51. The method ofclaim 49, wherein the number of layers of the data stream is one, andwherein the second codebook C₂ comprises at least one matrix of thefollowing form: $\begin{matrix}{{{W_{2} \in C_{2}} = \left\{ {{\frac{1}{\sqrt{2}}\begin{bmatrix}Y \\Y\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y \\{j\; Y}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y \\{- Y}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y \\{{- j}\; Y}\end{bmatrix}}} \right\}},{Y \in \left\{ {{\overset{\sim}{e}}_{1},{\overset{\sim}{e}}_{2},{\overset{\sim}{e}}_{3},{\overset{\sim}{e}}_{4}} \right\}}} & \;\end{matrix}$ where {tilde over (e)}₁ is a 4×1 vector with all zeroentries except for the i-th element which has value
 1. 52. The method ofclaim 49, wherein the number of layers of the data stream is one, andwherein the second codebook C₂ comprises at least one matrix of thefollowing form:${{W_{2} \in C_{2}} = \left\{ {{\frac{1}{2}\begin{bmatrix}Y \\{{\alpha (k)}Y}\end{bmatrix}},{\frac{1}{2}\begin{bmatrix}Y \\{j\; {\alpha (k)}Y}\end{bmatrix}},{\frac{1}{2}\begin{bmatrix}Y \\{{- {\alpha (k)}}Y}\end{bmatrix}},{\frac{1}{2}\begin{bmatrix}Y \\{{- j}\; {\alpha (k)}Y}\end{bmatrix}}} \right\}},\mspace{79mu} {Y \in \left\{ {{\overset{\sim}{e}}_{1},{\overset{\sim}{e}}_{2},{\overset{\sim}{e}}_{3},{\overset{\sim}{e}}_{4}} \right\}}$     α(k) = (^(j 2 π/32))^(2(k − 1)) where {tilde over (e)}₁ isa 4×1 vector with all zero entries except for the i-th element which hasvalue
 1. 53. The method of claim 49, wherein the number of layers of thedata stream is one, and wherein the second codebook C₂ comprises atleast one matrix of the following form:${{W_{2} \in C_{2}} = \left\{ {{\frac{1}{\sqrt{2}}\begin{bmatrix}Y \\{{\alpha (k)}Y}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y \\{j\; {\alpha (k)}Y}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y \\{{- {\alpha (k)}}Y}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y \\{{- j}\; {\alpha (k)}Y}\end{bmatrix}}} \right\}},\mspace{79mu} {Y \in \left\{ {{\overset{\sim}{e}}_{1},{\overset{\sim}{e}}_{2},{\overset{\sim}{e}}_{3},{\overset{\sim}{e}}_{4}} \right\}}$     α(k) = (^(j 2 π/32))^(2(k − 1)) where {tilde over (e)}₁ isa 4×1 vector with all zero entries except for the i-th element which hasvalue
 1. 54. The method of claim 49, wherein the number of layers of thedata stream is one, and wherein the precoding matrix W is chosen as afunction of the two PMI values i₁ and i₂ according to i₂ i₁ 0 1 2 3 40-15 W_(i) ₁ _(,0) ⁽¹⁾ W_(i) ₁ _(,8) ⁽¹⁾ W_(i) ₁ _(,16) ⁽¹⁾ W_(i) ₁_(,24) ⁽¹⁾ W_(i) ₁ _(+8,2) ⁽¹⁾ i₂ i₁ 5 6 7 8 9 0-15 W_(i) ₁ _(+8,10) ⁽¹⁾W_(i) ₁ _(+8,18) ⁽¹⁾ W_(i) ₁ _(+8,26) ⁽¹⁾ W_(i) ₁ _(+16,4) ⁽¹⁾ W_(i) ₁_(+16,12) ⁽¹⁾ i₂ i₁ 10 11 12 13 14 0-15 W_(i) ₁ _(+16,20) ⁽¹⁾ W_(i) ₁_(+16,28) ⁽¹⁾ W_(i) ₁ _(+24,6) ⁽¹⁾ W_(i) ₁ _(+24,14) ⁽¹⁾ W_(i) ₁_(+24,22) ⁽¹⁾ i₂ i₁ 15 0-15 W_(i) ₁ _(+24,30) ⁽¹⁾${{where}\mspace{14mu} W_{m,\; n}^{(1)}} = {\frac{1}{2}\begin{bmatrix}v_{m}^{\prime} \\{\phi_{n}^{\prime}v_{m}^{\prime}}\end{bmatrix}}$ φ′_(n) = e^(j2πn/32) v′_(m) = [1 e^(j2πm/32)]^(T).


55. The method of claim 49, wherein the number of layers of the datastream is two, and wherein the second codebook C₂ comprises at least onematrix of the following form:$\mspace{79mu} {{{W_{2} \in C_{2}} = \left\{ {{\frac{1}{2}\begin{bmatrix}Y_{1} & Y_{2} \\Y_{1} & {- Y_{2}}\end{bmatrix}},{\frac{1}{2}\begin{bmatrix}Y_{1} & Y_{2} \\{j\; Y_{1}} & {{- j}\; Y_{2}}\end{bmatrix}}} \right\}},{\left( {Y_{1},Y_{2}} \right) \in \left\{ {\left( {{\overset{\sim}{e}}_{1},{\overset{\sim}{e}}_{1}} \right),\left( {{\overset{\sim}{e}}_{2},{\overset{\sim}{e}}_{2}} \right),\left( {{\overset{\sim}{e}}_{3},{\overset{\sim}{e}}_{3}} \right),\left( {{\overset{\sim}{e}}_{4},{\overset{\sim}{e}}_{4}} \right),\left( {{\overset{\sim}{e}}_{1},{\overset{\sim}{e}}_{2}} \right),\left( {{\overset{\sim}{e}}_{2},{\overset{\sim}{e}}_{3}} \right),\left( {{\overset{\sim}{e}}_{1},{\overset{\sim}{e}}_{4}} \right),\left( {{\overset{\sim}{e}}_{2},{\overset{\sim}{e}}_{4}} \right)} \right\}}}$where {tilde over (e)}₁ is a 4×1 vector with all zero entries except forthe i-th element which has value
 1. 56. The method of claim 49, whereinthe number of layers of the data stream is two, and wherein the secondcodebook C₂ comprises at least one matrix of the following form:$\mspace{79mu} {{{W_{2} \in C_{2}} = \left\{ {{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\Y_{1} & {- Y_{2}}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\{j\; Y_{1}} & {{- j}\; Y_{2}}\end{bmatrix}}} \right\}},{\left( {Y_{1},Y_{2}} \right) \in \left\{ {\left( {{\overset{\sim}{e}}_{1},{\overset{\sim}{e}}_{1}} \right),\left( {{\overset{\sim}{e}}_{2},{\overset{\sim}{e}}_{2}} \right),\left( {{\overset{\sim}{e}}_{3},{\overset{\sim}{e}}_{3}} \right),\left( {{\overset{\sim}{e}}_{4},{\overset{\sim}{e}}_{4}} \right),\left( {{\overset{\sim}{e}}_{1},{\overset{\sim}{e}}_{2}} \right),\left( {{\overset{\sim}{e}}_{2},{\overset{\sim}{e}}_{3}} \right),\left( {{\overset{\sim}{e}}_{1},{\overset{\sim}{e}}_{4}} \right),\left( {{\overset{\sim}{e}}_{2},{\overset{\sim}{e}}_{4}} \right)} \right\}}}$where {tilde over (e)}₁ is a 4×1 vector with all zero entries except forthe i-th element which has value
 1. 57. The method of claim 49, whereinthe number of layers of the data stream is two, and wherein theprecoding matrix W is chosen as a function of the two PMI values i₁ andi₂ according to i₂ i₁ 0 1 2 3 0-15 W_(i) ₁ _(,i) ₁ _(,0) ⁽²⁾ W_(i) ₁_(,i) ₁ _(,1) ⁽²⁾ W_(i) ₁ _(+8,i) ₁ _(+8,0) ⁽²⁾ W_(i) ₁ _(+8,i) ₁_(+8,1) ⁽²⁾ i₂ i₁ 4 5 6 7 0-15 W_(i) ₁ _(+16,i) ₁ _(+16,0) ⁽²⁾ W_(i) ₁_(+16,i) ₁ _(+16,1) ⁽²⁾ W_(i) ₁ _(+24,i) ₁ _(+24,0) ⁽²⁾ W_(i) ₁ _(+24,i)₁ _(+24,1) ⁽²⁾ i₂ i₁ 8 9 10 11 0-15 W_(i) ₁ _(,i) ₁ _(+8,0) ⁽²⁾ W_(i) ₁_(,i) ₁ _(+8,1) ⁽²⁾ W_(i) ₁ _(+8,i) ₁ _(+16,0) ⁽²⁾ W_(i) ₁ _(+8,i) ₁_(+16,1) ⁽²⁾ i₂ i₁ 12 13 14 15 0-15 W_(i) ₁ _(,i) ₁ _(+24,0) ⁽²⁾ W_(i) ₁_(,i) ₁ _(+24,1) ⁽²⁾ W_(i) ₁ _(+8,i) ₁ _(+24,0) ⁽²⁾ W_(i) ₁ _(+8,i) ₁_(+24,1) ⁽²⁾${{where}\mspace{14mu} W_{m,m^{\prime},\; n}^{(2)}} = {\frac{1}{\sqrt{8}}\begin{bmatrix}v_{m}^{\prime} & v_{m^{\prime}}^{\prime} \\{\phi_{n}v_{m}^{\prime}} & {{- \phi_{n}}v_{m^{\prime}}^{\prime}}\end{bmatrix}}$ φ_(n) = e^(jπn/2) v′_(m) = [1 e^(j2πm/32)]^(T).


58. A method of channel state information (CSI) feedback in a wirelesscommunication system, node comprising: receiving one or more precodingmatrix indicator (PMI) signals from a remote transceiver, and generatinga precoding matrix W derived from a matrix multiplication of twomatrices W₁ and W₂, where said precoding matrix W is applicable toprecoding one or more layers of data streams, matrix W₁ is selected froma first codebook C₁ based on a first group of bits in the PMI signals,and matrix W₂ is selected from a second codebook C₂ based on a secondgroup of bits in the PMI signals, wherein the first codebook C₁ isconstructed asC₁ = {W₁⁽⁰⁾, W₁⁽¹⁾, …  , W₁⁽¹⁵⁾}  for  k = 0, 1, …  , 15 where${W_{1}^{(k)} = {{\begin{bmatrix}X^{(k)} & 0 \\0 & X^{(k)}\end{bmatrix}\mspace{14mu} {for}\mspace{14mu} k} = 0}},1,\ldots \mspace{14mu},15$usingX ^((k)) =[b _(k) b _((k+8)mod32) b _((k+10)mod32) b _((k+24)mod32)],b _(k)=[1e ^(j2πk/32)]^(T), for k=0,1, . . . ,31.
 59. The method ofclaim 58, wherein the number of layers of the data stream is one, andwherein the second codebook C₂ comprises at least one matrix of thefollowing form:${{W_{2} \in C_{2}} = \left\{ {{\frac{1}{2}\begin{bmatrix}Y \\Y\end{bmatrix}},{\frac{1}{2}\begin{bmatrix}Y \\{j\; Y}\end{bmatrix}},{\frac{1}{2}\begin{bmatrix}Y \\{- Y}\end{bmatrix}},{\frac{1}{2}\begin{bmatrix}Y \\{{- j}\; Y}\end{bmatrix}}} \right\}},$ for at least one vectorYε{{tilde over (e)} ₁ ,{tilde over (e)} ₂ ,{tilde over (e)} ₃ ,{tildeover (e)} ₄} where {tilde over (e)}_(i) is a 4×1 vector with all zeroentries except for the i-th element which has value
 1. 60. The method ofclaim 58, wherein the number of layers of the data stream is one, andwherein the second codebook C₂ comprises at least one matrix of thefollowing form:${{W_{2} \in C_{2}} = \left\{ {{\frac{1}{\sqrt{2}}\begin{bmatrix}Y \\Y\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y \\{j\; Y}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y \\{- Y}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y \\{{- j}\; Y}\end{bmatrix}}} \right\}},$ for at least one vectorYε{{tilde over (e)} ₁ ,{tilde over (e)} ₂ ,{tilde over (e)} ₃ ,{tildeover (e)} ₄} where {tilde over (e)}₁ is a 4×1 vector with all zeroentries except for the i-th element which has value
 1. 61. The method ofclaim 58, wherein the number of layers of the data stream is one, andwherein the second codebook C₂ comprises at least one matrix of thefollowing form:${{W_{2} \in C_{2}} = \left\{ {{\frac{1}{2}\begin{bmatrix}Y \\{{\alpha (k)}Y}\end{bmatrix}},{\frac{1}{2}\begin{bmatrix}Y \\{j\; {\alpha (k)}Y}\end{bmatrix}},{\frac{1}{2}\begin{bmatrix}Y \\{{- {\alpha (k)}}Y}\end{bmatrix}},{\frac{1}{2}\begin{bmatrix}Y \\{{- j}\; {\alpha (k)}Y}\end{bmatrix}}} \right\}},$ for at least one vectorYε{{tilde over (e)} ₁ ,{tilde over (e)} ₂ ,{tilde over (e)} ₃ ,{tildeover (e)} ₄} and at least one value k whereα(k)=(e ^(j2π/32))^(2(k-1)) and {tilde over (e)}₁ is a 4×1 vector withall zero entries except for the i-th element which has value
 1. 62. Themethod of claim 58, wherein the number of layers of the data stream isone, and wherein the second codebook C₂ comprises at least one matrix ofthe following form:${{W_{2} \in C_{2}} = \left\{ {{\frac{1}{\sqrt{2}}\begin{bmatrix}Y \\{{\alpha (k)}Y}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y \\{j\; {\alpha (k)}Y}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y \\{{- {\alpha (k)}}Y}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y \\{{- j}\; {\alpha (k)}Y}\end{bmatrix}}} \right\}},$ for at least one vectorYε{{tilde over (e)} ₁ ,{tilde over (e)} ₂ ,{tilde over (e)} ₃ ,{tildeover (e)} ₄} and at least one value k whereα(k)=(e ^(f2π/32))^(2(k-1)) and {tilde over (e)}₁ is a 4×1 vector withall zero entries except for the i-th element which has value
 1. 63. Themethod of claim 58, wherein the number of layers of the data stream isone, and wherein the precoding matrix W is chosen as a function of thetwo PMI values i₁ and i₂ according to i₂ i₁ 0 1 2 3 0-15 W_(i) ₁ _(,0)⁽¹⁾ W_(i) ₁ _(,8) ⁽¹⁾ W_(i) ₁ _(,16) ⁽¹⁾ W_(i) ₁ _(,24) ⁽¹⁾ i₂ i₁ 4 5 67 0-15 W_(i) ₁ _(+8,2) ⁽¹⁾ W_(i) ₁ _(+8,10) ⁽¹⁾ W_(i) ₁ _(+8,18) ⁽¹⁾W_(i) ₁ _(+8,26) ⁽¹⁾ i₂ i₁ 8 9 10 11 0-15 W_(i) ₁ _(+16,4) ⁽¹⁾ W_(i) ₁_(+16,12) ⁽¹⁾ W_(i) ₁ _(+16,20) ⁽¹⁾ W_(i) ₁ _(+16,28) ⁽¹⁾ i₂ i₁ 12 13 1415 0-15 W_(i) ₁ _(+24,6) ⁽¹⁾ W_(i) ₁ _(+24,14) ⁽¹⁾ W_(i) ₁ _(+24,22) ⁽¹⁾W_(i) ₁ _(+24,30) ⁽¹⁾${{where}\mspace{14mu} W_{m,n}^{(1)}} = {\frac{1}{2}\begin{bmatrix}v_{m}^{\prime} \\{\phi_{n}^{\prime}v_{m}^{\prime}}\end{bmatrix}}$ φ_(n) ^(′) = e^(j2πn/32) v_(m) ^(′) =[1 e^(j2πm/32)]^(T) where W = W_(m,n) ⁽¹⁾.

where W=W_(m,n) ⁽¹⁾.
 64. The method of claim 58, wherein the number oflayers of the data stream is two, and wherein the second codebook C₂comprises at least one matrix of the following form:${{W_{2} \in C_{2}} = \left\{ {{\frac{1}{2}\begin{bmatrix}Y_{1} & Y_{2} \\Y_{1} & {- Y_{2}}\end{bmatrix}},{\frac{1}{2}\begin{bmatrix}Y_{1} & Y_{2} \\{j\; Y_{1}} & {{- j}\; Y_{2}}\end{bmatrix}}} \right\}},$ for at least one pair of vectors satisfying(Y ₁ ,Y ₂)ε{({tilde over (e)} ₁ ,{tilde over (e)} ₁),({tilde over (e)} ₂,{tilde over (e)} ₂),({tilde over (e)} ₃ ,{tilde over (e)} ₃),({tildeover (e)} ₄ ,{tilde over (e)} ₄),({tilde over (e)} ₁ ,{tilde over (e)}₁),({tilde over (e)} ₂ ,{tilde over (e)} ₂),({tilde over (e)} ₃ ,{tildeover (e)} ₃),({tilde over (e)} ₄ ,{tilde over (e)} ₄)} where {tilde over(e)}₁ is a 4×1 vector with all zero entries except for the i-th elementwhich has value
 1. 65. The method of claim 58, wherein the number oflayers of the data stream is two, and wherein the second codebook C₂comprises at least one matrix of the following form:${{W_{2} \in C_{2}} = \left\{ {{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\Y_{1} & {- Y_{2}}\end{bmatrix}},{\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{1} & Y_{2} \\{j\; Y_{1}} & {{- j}\; Y_{2}}\end{bmatrix}}} \right\}},$ for at least one pair of vectors satisfying(Y ₁ ,Y ₂)ε{({tilde over (e)} ₁ ,{tilde over (e)} ₁),({tilde over (e)} ₂,{tilde over (e)} ₂),({tilde over (e)} ₃ ,{tilde over (e)} ₃),({tildeover (e)} ₄ ,{tilde over (e)} ₄),({tilde over (e)} ₁ ,{tilde over (e)}₁),({tilde over (e)} ₂ ,{tilde over (e)} ₂),({tilde over (e)} ₃ ,{tildeover (e)} ₃),({tilde over (e)} ₄ ,{tilde over (e)} ₄)} where {tilde over(e)}₁ is a 4×1 vector with all zero entries except for the i-th elementwhich has value
 1. 66. The method of claim 58, wherein the number oflayers of the data stream is two, and wherein the precoding matrix W ischosen as a function of the two PMI values i₁ and i₂ according to i₂ i₁0 1 2 3 0-15 W_(i) ₁ _(,i) ₁ _(,0) ⁽²⁾ W_(i) ₁ _(,i) ₁ _(,1) ⁽²⁾ W_(i) ₁_(+8,i) ₁ _(+8,0) ⁽²⁾ W_(i) ₁ _(+8,i) ₁ _(+8,1) ⁽²⁾ i₂ i₁ 4 5 6 0-15W_(i) ₁ _(+16,i) ₁ _(+16,0) ⁽²⁾ W_(i) ₁ _(+16,i) ₁ _(+16,1) ⁽²⁾ W_(i) ₁_(+24,i) ₁ _(+24,0) ⁽²⁾ i₂ i₁ 7 8 9 0-15 W_(i) ₁ _(+24,i) ₁ _(+24,1) ⁽²⁾W_(i) ₁ _(,i) ₁ _(+8,0) ⁽²⁾ W_(i) ₁ _(,j) ₁ _(+8,1) ⁽²⁾ i₂ i₁ 10 11 120-15 W_(i) ₁ _(+8,i) ₁ _(+16,0) ⁽²⁾ W_(i) ₁ _(+8,i) ₁ _(+16,1) ⁽²⁾ W_(i)₁ _(,i) ₁ _(+24,0) ⁽²⁾ i₂ i₁ 13 14 15 0-15 W_(i) ₁ _(,i) ₁ _(+24,1) ⁽²⁾W_(i) ₁ _(+8,i) ₁ _(+24,0) ⁽²⁾ W_(i) ₁ _(+8,i) ₁ _(+24,1) ⁽²⁾${{where}\mspace{14mu} W_{m,m^{\prime},n}^{(2)}} = {\frac{1}{\sqrt{8}}\begin{bmatrix}v_{m}^{\prime} & v_{m^{\prime}}^{\prime} \\{\phi_{n}v_{m}^{\prime}} & {{- \phi_{n}}v_{m^{\prime}}^{\prime}}\end{bmatrix}}$ φ_(n) = e^(jπn/2) v_(m) ^(′) = [1 e^(j2πm/32)]^(T) whereW = W_(m,m',n) ⁽²⁾.


67. A method of channel state information (CSI) feedback in a wirelesscommunication system, node comprising: receiving one or more precodingmatrix indicator (PMI) signals from a remote transceiver; and generatinga precoding matrix W derived from a matrix multiplication of twomatrices W₁ and W₂, where said precoding matrix W is applicable toprecoding one or more layers of data streams, matrix W₁ is selected froma first codebook using a PMI value i₁, and matrix W₂ is selected from asecond codebook using a PMI value i₂ where the first matrix is selectedaccording to $W_{1}^{(i_{1})} = \begin{bmatrix}X^{(i_{1})} & 0 \\0 & X^{(i_{1})}\end{bmatrix}$ for i₁ = 0, 1, …  , 15 with $X^{(k)} = \begin{bmatrix}b_{{kmod}\; 32} & b_{{({k + 8})}{mod}\; 32} & b_{{({k + 16})}{mod}\; 32} & b_{{({k + 24})}{mod}\; 32}\end{bmatrix}$ b_(k) = v_(k)^(′) $v_{k}^{\prime} = {\begin{bmatrix}1 & ^{{j\pi}\; {k/32}}\end{bmatrix}^{T}.}$
 68. The method of claim 67, wherein the number oflayers of the data stream is one, and wherein the second precoder matrixis selected according to $W_{2}^{(i_{2})} = {\frac{1}{2}\begin{bmatrix}{\overset{\sim}{e}}_{1 + {\lbrack{i_{2}/4}\rbrack}} \\{\phi_{{2{\lfloor{i_{2}/4}\rfloor}} + {8{({i_{2}\mspace{11mu} {mod}\mspace{14mu} 4})}}}^{\prime}{\overset{\sim}{e}}_{1 + {\lfloor{i_{2}/4}\rfloor}}}\end{bmatrix}}$ for i₂ = 0, 1, …  , 15 where ϕ_(n)^(′) = ^(jπ n/32)and {tilde over (e)}₁ is a 4×1 vector with all zero entries except forthe i-th element which has value
 1. 69. The method of claim 67, whereinthe number of layers of the data stream is one, and wherein theprecoding matrix W is chosen as a function of the two PMI values i₁ andi₂ according to i₂ i₁ 0 1 2 3 0-15 W_(i) ₁ _(,0) ⁽¹⁾ W_(i) ₁ _(,8) ⁽¹⁾W_(i) ₁ _(,16) ⁽¹⁾ W_(i) ₁ _(,24) ⁽¹⁾ i₂ i₁ 4 5 6 7 0-15 W_(i) ₁ _(+8,2)⁽¹⁾ W_(i) ₁ _(+8,10) ⁽¹⁾ W_(i) ₁ _(+8,18) ⁽¹⁾ W_(i) ₁ _(+8,26) ⁽¹⁾ i₂ i₁8 9 10 11 0-15 W_(i) ₁ _(+16,4) ⁽¹⁾ W_(i) ₁ _(+16,12) ⁽¹⁾ W_(i) ₁_(+16,20) ⁽¹⁾ W_(i) ₁ _(+16,28) ⁽¹⁾ i₂ i₁ 12 13 14 15 0-15 W_(i) ₁_(+24,6) ⁽¹⁾ W_(i) ₁ _(+24,14) ⁽¹⁾ W_(i) ₁ _(+24,22) ⁽¹⁾ W_(i) ₁_(+24,30) ⁽¹⁾${{where}\mspace{14mu} W_{m,n}^{(1)}} = {\frac{1}{2}\begin{bmatrix}v_{m}^{\prime} \\{\phi_{n}^{\prime}v_{m}^{\prime}}\end{bmatrix}}$ φ_(n) ^(′) = e^(j2πn/32) v_(m) ^(′) =[1 e^(j2πn/32)]^(T) where W = W_(m,n) ⁽¹⁾.


70. The method of claim 67, wherein the number of layers of the datastream is two, and wherein the second precoder matrix is selectedaccording to $W_{2}^{(i_{2})} = \left\{ \left\{ \left\{ \begin{matrix}{{\frac{1}{\sqrt{8}}\begin{bmatrix}{\overset{\sim}{e}}_{1 + {\lfloor{i_{2}/2}\rfloor}} & {\overset{\sim}{e}}_{1 + {\lfloor{i_{2}/2}\rfloor}} \\{^{{{j\pi}{({i_{2}\mspace{14mu} {mod}\mspace{14mu} 2})}}/2}{\overset{\sim}{e}}_{1 + {\lfloor{i_{2}/2}\rfloor}}} & {{- ^{{{j\pi}{({i_{2}\mspace{14mu} {mod}\mspace{14mu} 2})}}/2}}{\overset{\sim}{e}}_{1 + {\lfloor{i_{2}/2}\rfloor}}}\end{bmatrix}},} & {{{{if}\mspace{14mu} 0} \leq i_{2} \leq 7},} \\{{\frac{1}{\sqrt{8}}\begin{bmatrix}{\overset{\sim}{e}}_{1 + {\lfloor{{({i_{2} - 8})}/2}\rfloor}} & {\overset{\sim}{e}}_{2 + {\lfloor{{({i_{2} - 8})}/2}\rfloor}} \\{^{{{j\pi}{({i_{2}\mspace{14mu} {mod}\mspace{14mu} 2})}}/2}{\overset{\sim}{e}}_{1 + {\lfloor{i_{2} - {8/2}}\rfloor}}} & {{- ^{{{j\pi}{({i_{2}\mspace{14mu} {mod}\mspace{14mu} 2})}}/2}}{\overset{\sim}{e}}_{2 + {\lfloor{{({i_{2} - 8})}/2}\rfloor}}}\end{bmatrix}},} & {{{{if}\mspace{14mu} 8} \leq i_{2} \leq 11},} \\{{\frac{1}{\sqrt{8}}\begin{bmatrix}{\overset{\sim}{e}}_{1 + {\lfloor{{({i_{2} - 12})}/2}\rfloor}} & {\overset{\sim}{e}}_{4} \\{^{{{j\pi}{({i_{2}\mspace{14mu} {mod}\mspace{14mu} 2})}}/2}{\overset{\sim}{e}}_{1 + {\lfloor{{({i_{2} - 12})}/2}\rfloor}}} & {{- ^{{{j\pi}{({i_{2}\mspace{14mu} {mod}\mspace{14mu} 2})}}/2}}{\overset{\sim}{e}}_{4}}\end{bmatrix}},} & {{{if}\mspace{14mu} 12} \leq i_{2} \leq 15}\end{matrix} \right. \right. \right.$ where {tilde over (e)}₁ is a 4×1vector with all zero entries except for the i-th element which hasvalue
 1. 71. The method of claim 67, wherein the number of layers of thedata stream is two, and wherein the precoding matrix W is chosen as afunction of the two PMI values i₁ and i₂ according to i₂ i₁ 0 1 2 3 0-15W_(i) ₁ _(,i) ₁ _(,0) ⁽²⁾ W_(i) ₁ _(,i) ₁ _(,1) ⁽²⁾ W_(i) ₁ _(+8,i) ₁_(+8,0) ⁽²⁾ W_(i) ₁ _(+8,i) ₁ _(+8,1) ⁽²⁾ i₂ i₁ 4 5 6 0-15 W_(i) ₁_(+16,i) ₁ _(+16,0) ⁽²⁾ W_(i) ₁ _(+16,i) ₁ _(+16,1) ⁽²⁾ W_(i) ₁ _(+24,i)₁ _(+24,0) ⁽²⁾ i₂ i₁ 7 8 9 0-15 W_(i) ₁ _(+24,i) ₁ _(+24,1) ⁽²⁾ W_(i) ₁_(,i) ₁ _(+8.0) ⁽²⁾ W_(i) ₁ _(,i) ₁ _(+8,1) ⁽²⁾ i₂ i₁ 10 11 12 0-15W_(i) ₁ _(+8,i) ₁ _(+16,0) ⁽²⁾ W_(i) ₁ _(+8,i) ₁ _(+16,1) ⁽²⁾ W_(i) ₁_(,i) ₁ _(+24,0) ⁽²⁾ i₂ i₁ 13 14 15 0-15 W_(i) ₁ _(,i) ₁ _(+24,1) ⁽²⁾W_(i) ₁ _(+8,i) ₁ _(+24,0) ⁽²⁾ W_(i) ₁ _(+8,i) ₁ _(+24,1) ⁽²⁾${{where}\mspace{14mu} W_{m,m^{\prime},n}^{(2)}} = {\frac{1}{\sqrt{8}}\begin{bmatrix}v_{m}^{\prime} & v_{m^{\prime}}^{\prime} \\{\phi_{n}v_{m}^{\prime}} & {{- \phi_{n}}v_{m^{\prime}}^{\prime}}\end{bmatrix}}$ φ_(n) = e^(jπn/2) v_(m) ^(′) = [1 e^(j2πn/32)]^(T) whereW = W_(m,m',n) ⁽²⁾.